Gradient and Intercept

Consider a linear function whose graph is a straight line. But how can we concretely plot the straight line in order to determine its properties? 

Gradient and Intercept Gradient and Intercept

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    Gradients and intercepts are the only piece of information we need in order to plot the graph of a straight line and know its properties. Let us explore what they are in this article.

    Gradient and intercepts of a line meaning

    Gradient of a line meaning

    Suppose a man is walking up a hill, and for keeping things simple, the hill is straight in its path with no bumps and curves.

    If the man travels 100 meters in the horizontal direction and finds himself 200 meters above the ground, how can we measure the steepness of the hill that is, how inclined the hill is?

    Gradient and Intercepts, A man standing on top of a hill, StudySmarter A man standing on top of a hill, StudySmarter Originals

    To be more rigorous, the sloppiness, in mathematical terms, is known as the Gradient of the hill.

    The gradient of a straight line is the measure of how steep it is, or how sloppy it is.

    Intercepts of a line meaning

    A linear function of the form ax+by+c=0 is represented by a straight line that extends infinitely in either direction. Hence, at one point it will intersect either the y-axis or the x-axis or more commonly, both the axes of the coordinate plane.

    The coordinates of the point where a straight line intersects either of the axes is known as the intercept of that straight line.

    Types of intercepts of a line

    And as there are two axes in the coordinate plane, a straight line may have intersections with the x-axis or with the y-axis, or with both of them.

    The x-intercepts are the points where a straight line intersects the x-axis.

    The y-intercepts are the points where a straight line intersects the y-axis.

    Gradient and intercepts calculation and formula

    Gradient calculation and formula

    The gradient of a line is the ratio of the change in the y-direction to the change in the x-direction.

    To measure the steepness of the hill, an intuitive way is to know how the change in the vertical direction is with respect to the change in a horizontal direction, which is essentially what we want to know. And this is exactly how the gradient is defined mathematically.

    Translating this definition into an equation, with m denoting the gradient, we get

    m=yx=yf-yixf-xi

    where yf, yi denote the final and initial y-coordinates, and xf, xidenote the final and initial x-coordinates.

    Geometrical interpretation of the gradient

    On close inspection, it can be observed that the formula of the gradient which is equal to the ratio between the change in y over the change in x, is nothing but the tangent of the angle formed between the line and the positive x-axis.

    m=tan θ

    Now if we go back to our man on the top of the hill, we can calculate the gradient of the hill with our formula,

    m=200-0100-0=200100=2

    Hence, the hill has a gradient of 2.

    But how can we find the gradient of a line if we are given nothing but only the equation of the line? According to the definition of the gradient, we recall

    m=yx=yf-yixf-xi.

    Hence we need the coordinates of any two points on the line to calculate the gradient.

    But there are infinite of them to choose from, and how do we know which ones to choose? Recall that we do know the coordinates of two specific points on the line, the x and y-intercepts!

    So we can choose the two points as xi,yi=-ca,0 and xf,yf=0,-cb. After substitution we get,

    m=-cb-00--ca=-cbca=-cb×ac=-ab

    Therefore, the gradient (or slope) of a line given by 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is m=-ab{"x":[[128,127,126,126,125,125,125,123,123,122,121,120,118,117,116,116,115,114,116,118,121,125,127,136,141,152,156,160,167,170,177,180,182,183,192,192,193,193,194,195,197,198,202,204,211,216,224,227,231,238,245,247,254,258,260,260,261,261,260,258,257,256,256,256,258,260],[347,347,347,349,354,359,366,369,375,385,387],[345,352,354,357,361,370,375,380,389,394,398,406],[515,516,518,521,529,539,549,555,560,569,574,578,587,594,600,605,609,612],[670,673,676,679,684,696,709,716,738,745,767,775,794,807,817,821,831,833],[735,738,739,740,742,742,742,742,741,739,737,732,727,722,720,712,710,707,707,707,709,712,715,717,719,723,726,728,733,735,739,745,749,751,756,757,758,761,764,767,769,771,775,777,780,782,785],[732,732,732,732,733,733,734,734,733,732,730,729,728,727,727,727,727,727,728,729,732,733,734,738,741,746,752,758,764,770,773,779,781,782,783,783,780,778,770,763,759,749,746,738,734,732,731]],"y":[[239,240,241,242,248,265,273,290,295,300,310,314,324,327,329,331,335,335,331,324,316,306,301,284,278,265,263,262,261,261,266,269,271,274,293,296,306,310,313,314,313,311,303,299,285,273,260,255,250,240,234,232,230,231,235,237,244,256,261,276,281,294,298,307,311,313],[197,196,195,195,195,196,196,197,198,198,198],[242,242,241,240,239,237,236,235,234,233,232,230],[225,225,225,225,223,222,220,219,218,216,216,215,214,213,212,212,212,212],[282,281,280,279,278,276,274,272,270,269,268,267,266,266,266,266,266,267],[180.5,177.5,176.5,175.5,173.5,171.5,169.5,166.5,164.5,162.5,162.5,161.5,163.5,167.5,170.5,182.5,187.5,201.5,208.5,210.5,214.5,215.5,214.5,213.5,212.5,208.5,206.5,203.5,197.5,194.5,188.5,182.5,178.5,178.5,178.5,179.5,181.5,186.5,191.5,195.5,197.5,199.5,201.5,202.5,202.5,202.5,201.5],[305.5,302.5,303.5,304.5,309.5,312.5,321.5,327.5,346.5,352.5,369.5,374.5,378.5,386.5,390.5,398.5,400.5,404.5,405.5,405.5,405.5,404.5,403.5,398.5,396.5,391.5,387.5,383.5,382.5,382.5,382.5,384.5,385.5,387.5,390.5,394.5,399.5,402.5,410.5,416.5,418.5,424.5,425.5,425.5,423.5,418.5,414.5]],"t":[[0,5,10,17,24,32,41,49,57,58,66,74,86,91,91,102,107,119,141,152,157,169,174,185,191,202,207,208,219,224,236,241,241,252,278,285,291,302,307,325,332,341,352,357,368,374,385,391,391,402,407,419,424,435,441,452,457,469,474,486,491,502,508,519,524,535],[778,787,799,808,818,824,835,841,852,858,868],[1052,1065,1069,1074,1076,1085,1091,1091,1102,1108,1108,1118],[1511,1533,1541,1544,1552,1558,1569,1574,1575,1583,1591,1592,1602,1608,1621,1625,1637,1641],[2654,2671,2675,2683,2683,2691,2702,2708,2719,2725,2736,2742,2753,2758,2769,2775,2786,2791],[1649599989432,1649599989452,1649599989456,1649599989464,1649599989472,1649599989484,1649599989489,1649599989500,1649599989508,1649599989514,1649599989523,1649599989534,1649599989540,1649599989551,1649599989556,1649599989567,1649599989573,1649599989584,1649599989589,1649599989600,1649599989606,1649599989618,1649599989623,1649599989623,1649599989634,1649599989639,1649599989640,1649599989651,1649599989656,1649599989656,1649599989667,1649599989673,1649599989684,1649599989689,1649599989700,1649599989706,1649599989710,1649599989714,1649599989723,1649599989734,1649599989739,1649599989740,1649599989748,1649599989756,1649599989757,1649599989767,1649599989770],[1649599990213,1649599990232,1649599990241,1649599990251,1649599990256,1649599990256,1649599990268,1649599990273,1649599990285,1649599990290,1649599990301,1649599990306,1649599990309,1649599990314,1649599990323,1649599990334,1649599990339,1649599990348,1649599990356,1649599990368,1649599990373,1649599990373,1649599990385,1649599990389,1649599990390,1649599990401,1649599990406,1649599990418,1649599990423,1649599990434,1649599990440,1649599990451,1649599990456,1649599990458,1649599990468,1649599990473,1649599990485,1649599990490,1649599990501,1649599990506,1649599990518,1649599990523,1649599990534,1649599990539,1649599990551,1649599990556,1649599990568]],"version":"2.0.0"}.

    Remember that slope and gradient share the same meaning in this context of a straight line.

    Intercepts calculation

    We consider the graph of a straight line intercepting the axes of the coordinate plane.

    Gradient and Intercepts, The graph of a linear function ax+by+c=0, StudySmarterThe graph of a linear function ax+by+c=0, StudySmarter Originals

    We notice that as per the definition, the intercepts are formed on the axes themselves.

    The x-intercept will lie on the x-axis and so the y coordinate will be 0 at that point.

    • To calculate the x-intercept, set y=0 and solve for x.

    Consider a straight line, modeled by the linear equation, where a, b and c are real-valued constants, with a and b not being simultaneously 0.

    To find the x-intercepts of this line, we need to substitute y by 0 since the y-coordinate vanishes at that point. Hence, we get

    ax+b0+c=0 ax+c=0 ax=-c x=-ca

    Thus, the x-intercept of the line ax+by+c=0, is -ca,0.

    • To calculate the y-intercept, set x=0 and solve for y.

    To find the y-intercept of the line, we need to substitute x by 0, which yields

    a(0)+by+c=0 by+c=0 by=-c y=-cb

    Thus, the y-intercept of the line ax+by+c=0 is 0,-cb.

    Gradient and Intercepts examples

    Find the x-intercept and the y-intercept of the line of equation2x+4y-1=0{"x":[[95,94,93,92,92,91,91,91,92,93,96,100,104,110,114,119,121,123,127,128,130,132,134,134,133,131,128,120,110,105,96,94,88,85,85,85,86,89,95,105,109,117,131,141,156,165,176,185,191],[221,221,220,221,225,227,234,237,245,249,257,259,261,264,268,270,272,275,276,276,276,276,275,274,271,266,263,256,253,247,243,240,238],[311,309,306,304,302,299,296,294,289,287,284,282,281,281,281,282,284,286,289,291,293,298,301,306,313,320,323,330],[365,364,363,362,364,367,369,377,380,390,395,407,414,421,425,427,432,436,438],[406,406,406,407,408,408,408,404,399,397,396,394,394,394,395,397],[482,484,484,485,485,485,482,477,473,468,467,465,463,462,461,461,462,463,469,473,482,489,495,507,513,524,531,537,540,545,550,555],[504,504,504,504,504,504,504,503,503,501,500,499,498,498,498,498,498],[577,576,576,576,576,575,576,577,579,582,585,591,596,605,610,613,616,620,624,626,628,635,637,639,642,643,643,643,642,641,640,639,637,636,635,634,633,631,629,626,621,620,618,617,614,609,606,598,596,593,589,585,584],[686,688,691,694,703,706,714,722,725,729,735,741,747,752,756,758,761,762],[816,816,816,816,815,815,813,812,810,808,808,807,806,806,806,805,805,805,805,805,806,806,807,808,809],[886,885,887,890,892,894,900,904,907,916,919,923,930,933,936,941,943,944,947],[872,870,871,876,880,887,895,903,907,919,922,933,935,940,944,946,948,950,951,952],[1003,1002,1001,998,995,992,990,989,988,988,988,990,991,996,1000,1004,1008,1011,1018,1021,1028,1030,1037,1040,1047,1048,1051,1054,1055,1056,1056,1055,1053,1051,1047,1041,1034,1030,1018,1015,1006,1004,1000,1000,1001,1002]],"y":[[322,320,318,317,315,311,310,305,303,300,295,291,288,286,285,286,286,288,291,293,296,305,316,319,328,338,349,365,381,390,402,405,414,418,419,422,422,422,421,419,418,415,412,410,407,405,402,399,397],[327,326,323,320,315,313,308,307,306,306,308,309,310,312,315,317,320,325,328,331,337,341,345,349,357,366,370,381,384,393,397,399,401],[301,302,305,306,308,312,318,321,330,334,345,352,359,362,366,371,376,378,382,383,385,387,388,388,388,387,386,383],[351,351,351,351,350,350,350,349,349,347,347,345,344,343,343,342,342,341,341],[317,316,315,315,318,319,327,350,368,376,382,390,395,396,398,398],[284,284,285,288,290,294,305,315,326,339,341,349,354,355,358,361,364,365,365,365,365,364,363,362,361,360,360,360,360,361,361,362],[336,335,336,338,346,354,369,375,379,391,398,406,415,416,417,418,419],[333,334,336,339,342,354,361,368,375,378,380,381,381,377,373,371,369,364,359,355,352,339,335,331,327,326,329,333,340,352,361,366,384,389,395,401,407,418,428,437,452,455,459,461,465,468,468,463,461,456,449,439,436],[342,342,342,342,342,342,342,341,341,340,340,340,340,340,340,341,342,342],[282,281,280,284,287,291,302,315,329,342,347,353,362,367,371,378,385,390,394,398,400,401,402,401,400],[307,307,307,307,307,306,306,306,306,305,305,305,305,305,305,305,305,305,305],[346,349,350,350,350,349,348,346,345,343,343,341,340,339,339,338,338,338,338,338],[299,301,302,309,319,330,342,348,359,371,379,383,386,395,398,401,402,402,401,400,394,391,381,377,362,357,346,335,325,316,312,308,302,300,296,293,292,291,294,295,300,302,307,308,314,315]],"t":[[0,3,13,24,26,37,43,54,62,76,78,93,104,110,120,127,137,140,143,154,163,175,177,188,193,204,210,221,237,243,251,263,273,278,289,293,310,321,326,337,346,354,363,374,389,393,404,421,425],[721,729,735,743,754,765,774,777,788,794,803,810,810,820,826,827,837,843,844,854,860,860,868,871,877,888,893,904,910,920,926,937,943],[1169,1175,1180,1185,1187,1193,1204,1210,1221,1227,1235,1243,1254,1260,1260,1271,1281,1289,1293,1294,1304,1310,1310,1320,1327,1337,1344,1354],[1727,1735,1744,1752,1780,1788,1794,1804,1811,1820,1827,1837,1843,1854,1860,1861,1871,1877,1885],[2071,2086,2089,2101,2119,2127,2138,2154,2171,2187,2194,2204,2212,2221,2227,2239],[2429,2441,2452,2460,2471,2477,2488,2504,2511,2527,2538,2544,2554,2560,2570,2577,2588,2596,2607,2619,2627,2638,2652,2660,2671,2677,2685,2694,2705,2711,2721,2727],[3069,3084,3107,3111,3127,3137,3144,3154,3161,3171,3177,3188,3205,3213,3214,3222,3227],[3740,3769,3780,3782,3789,3794,3805,3822,3828,3839,3844,3855,3861,3878,3888,3897,3899,3905,3911,3920,3928,3938,3946,3955,3961,3972,3994,4005,4011,4024,4028,4036,4044,4045,4055,4062,4065,4072,4078,4089,4103,4106,4111,4122,4128,4141,4144,4161,4172,4178,4188,4194,4205],[5964,6013,6022,6030,6045,6045,6056,6062,6062,6073,6078,6089,6095,6110,6112,6113,6128,6139],[6559,6572,6575,6595,6604,6604,6615,6626,6629,6639,6645,6646,6657,6662,6662,6673,6679,6690,6695,6706,6712,6723,6729,6740,6742],[6984,7000,7012,7021,7029,7030,7040,7046,7049,7057,7062,7062,7074,7079,7079,7090,7095,7096,7104],[7258,7271,7279,7290,7297,7307,7314,7323,7330,7340,7345,7357,7362,7374,7379,7379,7390,7396,7396,7407],[7639,7640,7646,7654,7662,7673,7679,7690,7696,7707,7712,7713,7721,7729,7740,7747,7757,7763,7773,7779,7790,7797,7807,7812,7824,7829,7840,7846,7857,7862,7863,7874,7879,7880,7891,7896,7907,7912,7924,7930,7940,7946,7957,7962,7974,7979]],"version":"2.0.0"}.

    Solution

    Comparing the given equation of the line with the general form: ax+by+c=0{"x":[[162,164,164,164,163,161,157,149,142,132,123,113,104,101,92,91,89,90,96,102,109,117,121,134,138,143,151,155,162,165,169,171,172,172,172,171,171,171,171,173,174],[214,214,212,213,215,223,230,234,245,248,255,257,259,259,257,254,252,244,241,234,232,229,229],[296,289,288,285,281,279,272,263,260,258,257,257,257,262,265,277,287,298],[353,351,351,352,353,357,360,362,369,380,388,394,397,407,411,414,416,417],[379,376,376,376,378,378,379,381,381,383,384,384],[477,477,477,475,474,467,463,454,449,445,442,441,440,440,441,443,446,449,454,462,470,474,485,489,495,498,499,499,499,497,493,491,483,480,476,469,461,452,446,441,440,440],[525,519,519,519,518,516,516,516,517],[527,529,533,538,543,549,555,557,564,568,569,568,566,565,563,558,557,553,552,549,547,545,542,540,539,538,538,542,544],[612,612,612,614,619,626,634,638,651,655,661,664,669,673,675,676],[637,637,639,640,641,643,643,643,642,641,641],[749,749,749,748,748,745,744,737,734,730,722,714,706,701,699,695,695,698,702,708,715,720,733],[821,822,824,828,831,843,846,857,863,869,872,874],[814,820,823,839,849,859,868],[940,944,946,949,950,949,945,943,935,933,930,928,924,923,921,921,921,924,928,931,942,945,956,960,970,973,981,984,985,985,985,981,979,972,964,955,946,939,934]],"y":[[271,253,249,242,239,234,231,230,235,243,253,265,279,285,304,310,322,324,325,323,318,310,306,293,288,282,273,269,264,263,265,271,280,291,304,311,330,335,339,345,347],[268,267,255,253,251,245,241,240,239,239,245,248,260,269,278,288,293,308,313,325,328,333,334],[244,241,241,245,251,256,273,297,308,313,322,326,329,334,335,335,334,330],[284,280,279,279,278,277,277,277,276,277,277,278,278,277,276,275,273,272],[256,255,254,255,261,264,267,276,281,290,300,304],[188,185,186,191,195,213,221,247,261,274,286,290,299,300,301,300,298,296,291,284,277,274,269,269,272,277,283,286,289,295,300,302,306,307,308,309,310,310,310,307,304,301],[251,250,252,254,261,270,275,287,290],[295,294,291,287,282,276,270,266,257,253,255,257,265,271,277,298,306,323,331,345,357,369,380,391,397,407,409,411,409],[277,276,275,274,274,274,275,276,277,277,277,276,275,274,273,272],[258,257,258,259,261,268,278,284,300,305,308],[248,245,242,239,238,235,235,235,236,238,242,250,259,268,274,290,295,306,312,316,319,320,320],[249,246,246,246,246,245,245,244,244,243,243,242],[286,286,285,281,278,276,273],[233,227,225,220,220,221,225,228,240,245,251,256,269,276,290,301,306,317,320,321,320,318,311,308,296,291,274,261,248,233,227,217,212,206,206,208,213,220,228]],"t":[[0,18,22,26,37,43,51,60,70,76,87,93,103,109,120,126,137,143,154,159,169,176,187,193,202,206,210,220,226,227,237,243,254,260,270,276,289,293,293,304,309],[503,523,526,527,535,543,554,560,570,576,587,593,601,610,618,626,635,643,654,660,668,676,687],[830,849,852,860,869,876,887,904,910,921,927,927,937,943,954,960,971,977],[1217,1236,1239,1243,1254,1260,1260,1271,1277,1287,1294,1304,1310,1326,1327,1335,1343,1354],[1467,1474,1478,1493,1504,1510,1510,1521,1527,1537,1543,1550],[1718,1734,1744,1754,1760,1771,1777,1787,1793,1804,1810,1823,1827,1838,1843,1852,1860,1861,1871,1877,1888,1893,1904,1910,1923,1927,1938,1943,1944,1955,1961,1971,1977,1988,1990,1994,2006,2010,2021,2027,2036,2044],[2232,2249,2260,2261,2270,2277,2288,2294,2305],[2327,2330,2336,2344,2354,2360,2371,2378,2388,2404,2410,2421,2427,2427,2438,2444,2455,2460,2461,2472,2477,2488,2494,2505,2510,2521,2527,2538,2544],[2814,2822,2831,2836,2844,2855,2861,2871,2877,2888,2894,2894,2908,2911,2922,2927],[3052,3061,3077,3086,3086,3094,3105,3111,3122,3127,3132],[3379,3386,3394,3405,3411,3422,3428,3438,3444,3445,3455,3461,3472,3477,3492,3496,3505,3511,3522,3527,3539,3544,3552],[3827,3844,3847,3853,3861,3872,3877,3889,3894,3905,3911,3911],[4059,4078,4086,4094,4106,4111,4122],[4284,4289,4296,4305,4311,4330,4341,4345,4353,4361,4365,4372,4378,4378,4389,4394,4406,4411,4422,4428,4439,4444,4456,4461,4472,4478,4489,4494,4506,4511,4522,4528,4528,4540,4544,4556,4561,4572,4578]],"version":"2.0.0"},

    We get a=2, b=4, c=-1.

    And we know that the y-intercept is given by

    -cb=--14{"x":[[151,152,157,163,172,181,186,200,204,218,221,232,237,239,241,242],[379,378,376,374,373,368,366,357,353,341,333,329,316,312,302,299,294,292,291,291,294,298,305,309,322,327,344,355,367,371],[292,290,292,296,302,310,320,331,342,352,362,370,377,382,384,386],[347,345,344,343,342,342,340,339,337,335,334,333,332,332,333,336,339,343,348,351,359,361,366,368,372,375,377,377,377,377,373,371,365,363,360,355,351,346,343,342],[495,491,489,486,483,481],[477,478,480,484,486,489,497,504,512,520,527,532,537,538,541,541,539],[471,470,472,474,485,495,508,520,531,540,545,555,559,563,564],[628,631,635,642,651,660,663,667,675,680,683],[795,794,793,792,791,788,786,779,776,768,764,760,751,746,737,729,723,720,714,712,710,710,714,717,720,727,735,744,748,753,763,773,778],[801,807,809,812,823,827,831,839,846,849,859,861,867],[917,920,920,921,921,922,921,919,917,915,913,911,910,910,910,911,913,913],[831,839,847,858,870,876,882,894,899,905,916,921,925,934,940,945,947,949],[882,882,881,879,876,873,872,870,868,866,865,865,865,867,869,873,878,884,887,898,901,911,914,916,921,923,925,927,930,931],[900,899,899,899,898,896,895,893,892,891,891,890,891,893,894,895,897],[995,995,995,997,1001,1004,1007,1011,1013,1018,1020,1023,1024,1026,1029,1030,1030,1028,1026,1024,1019,1013]],"y":[[294,294,293,291,288,286,285,282,281,278,277,276,276,276,277,277],[149,147,142,138,137,134,134,134,135,140,144,147,159,164,179,184,195,200,208,215,220,224,228,229,230,230,228,225,222,221],[288,289,289,288,287,285,282,279,276,274,272,271,271,270,271,272],[308,309,311,323,329,334,347,359,369,379,388,395,397,403,404,401,396,391,385,383,377,376,375,376,378,382,387,389,397,399,407,409,414,415,417,418,418,418,415,409],[244,245,246,248,250,251],[249,249,249,249,249,249,249,248,247,245,244,243,243,243,243,244,245],[284,284,284,284,283,281,279,277,275,273,272,270,270,270,270],[260,260,259,258,257,255,254,253,251,250,250],[145,142,139,138,138,137,137,141,143,150,154,159,171,178,194,212,233,244,279,291,328,350,369,377,383,393,400,406,408,409,410,409,407],[221,219,219,218,217,217,217,216,215,214,212,212,210],[142,135,134,134,135,138,143,150,158,167,172,187,191,201,204,208,210,211],[281,280,278,276,275,274,273,272,271,271,269,269,269,268,268,268,268,269],[302,303,311,321,330,339,343,349,354,359,360,362,364,365,365,365,365,363,362,360,359,356,356,355,355,354,354,354,354,353],[333,331,332,335,341,348,356,365,370,379,382,387,388,388,388,388,387],[133,125,123,120,121,125,132,142,149,171,180,199,207,215,241,259,277,296,307,316,335,354]],"t":[[0,7,22,33,38,50,55,66,71,83,88,99,105,116,121,138],[477,482,490,499,505,516,522,532,538,549,555,566,572,582,588,599,605,605,615,622,632,639,649,657,665,672,682,688,699,705],[958,976,982,988,999,1005,1016,1023,1032,1038,1049,1055,1063,1072,1082,1102],[1453,1472,1480,1489,1497,1500,1505,1516,1522,1533,1539,1549,1555,1566,1583,1598,1605,1616,1622,1633,1639,1650,1655,1656,1668,1672,1683,1689,1699,1705,1718,1722,1733,1739,1739,1750,1755,1766,1772,1783],[1957,1977,1983,1989,1999,2006],[2479,2498,2506,2516,2522,2523,2533,2539,2550,2556,2567,2573,2583,2589,2600,2606,2614],[2813,2830,2839,2850,2856,2866,2873,2883,2889,2900,2906,2917,2922,2933,2939],[4083,4116,4124,4135,4140,4148,4156,4157,4167,4174,4184],[4569,4574,4582,4590,4601,4606,4617,4623,4632,4640,4640,4651,4656,4657,4668,4673,4684,4690,4701,4707,4718,4723,4734,4740,4740,4751,4756,4767,4773,4774,4786,4790,4798],[5205,5221,5224,5232,5240,5248,5249,5257,5267,5273,5284,5290,5301],[5493,5504,5508,5515,5524,5534,5540,5551,5557,5568,5574,5584,5590,5601,5607,5618,5623,5631],[6001,6011,6017,6024,6035,6040,6043,6052,6057,6058,6068,6074,6074,6085,6090,6101,6107,6118],[6340,6357,6368,6374,6382,6390,6401,6407,6418,6424,6424,6435,6440,6452,6457,6468,6474,6485,6491,6502,6507,6518,6524,6524,6535,6540,6541,6552,6557,6568],[6709,6727,6741,6750,6757,6768,6775,6785,6791,6802,6807,6816,6824,6835,6841,6841,6852],[7097,7107,7111,7116,7135,7141,7152,7157,7169,7174,7185,7191,7191,7202,7207,7219,7224,7235,7241,7242,7252,7257]],"version":"2.0.0"}=14{"x":[[202,206,209,215,224,233,241,246,261,266,281,290,298,319],[209,211,215,222,226,231,239,243,252,261,270,275,284,288,300,303,307],[489,489,489,490,491,492,493,493,493,491,487,484,481,479,478,478,477,477],[418,422,429,444,449,464,469,484,489,505,509,521,528,531,537,540],[472,472,472,471,467,463,457,454,447,444,439,436,435,435,436,442,444,454,462,471,480,490,494,498,502,514,517,526,530,532,533,533],[499,499,499,499,499,499,498,498,494,492,490,490,489,489,491,492]],"y":[[261,261,260,259,257,256,255,254,253,252,251,251,251,252],[290,290,289,288,287,286,285,284,282,280,278,278,276,276,275,275,275],[143,142,140,138,137,137,141,144,147,156,172,184,194,202,208,211,217,219],[286,286,286,286,286,286,286,285,285,284,284,283,284,284,286,288],[309,316,321,325,336,345,355,359,373,377,388,394,398,401,402,405,405,405,404,402,401,400,399,399,399,399,399,400,401,402,404,405],[356,355,354,353,354,355,360,364,379,392,406,411,421,429,438,443]],"t":[[0,9,19,25,35,40,52,57,68,74,85,90,103,130],[371,407,418,424,432,436,441,449,457,468,474,474,485,491,501,507,512],[846,849,853,857,868,885,891,891,901,907,918,924,935,941,951,957,968,974],[1219,1228,1237,1249,1257,1268,1274,1285,1291,1299,1307,1318,1324,1333,1341,1352],[1569,1586,1593,1599,1608,1620,1624,1633,1641,1650,1658,1667,1674,1685,1691,1702,1708,1716,1724,1735,1741,1749,1758,1758,1768,1774,1785,1791,1802,1809,1816,1827],[1988,1992,2004,2008,2024,2025,2035,2041,2052,2058,2069,2074,2085,2091,2102,2108]],"version":"2.0.0"}.

    Hence the y-intercept lies at 0,14{"x":[[239,219,216,213,206,196,190,185,181,178,177,178,179,185,188,198,202,211,226,232,272],[329,326,321,317,315,308,307,302,301,301,301,304,308,313,319,322,332,336,344,347,355,359,370,373,382,384,387,387,385,382,376,362,354,350,344,341,336,333,332],[449,448,447,447,447,447,447,447,446,446,446],[560,562,563,564,564,562,561,559,556,555,552],[528,527,528,533,536,552,558,577,584,604,617,630,641,646,650,662,664,665],[584,584,585,586,586,585,584,580,579,577,574,571,570,569,569,573,574,582,586,593,598,602,615,625,633,641,644,651,654,658],[614,613,613,613,613,612,611,608,608,606,604,604,604,604],[721,723,733,737,742,744,748,751,754,756,758,760,760,760,757,755,748,744,732,728,714,704]],"y":[[178,209,217,225,243,275,300,324,350,376,388,423,434,463,472,495,502,514,528,531,539],[246,248,253,263,269,293,301,327,344,359,374,387,397,404,408,408,407,405,399,395,386,380,361,354,329,322,297,288,267,256,247,239,239,239,241,243,248,252,255],[377,376,375,374,376,378,388,392,406,411,418],[193,184,183,188,191,203,208,214,226,232,244],[339,339,337,337,337,336,335,333,332,328,326,324,323,323,322,323,324,324],[353,354,357,359,362,373,378,392,397,402,411,420,423,433,436,441,442,445,445,446,446,446,446,446,445,445,445,445,446,446],[409,409,411,416,424,435,440,457,462,471,479,484,487,492],[192,188,177,182,189,194,206,222,239,259,281,314,326,362,387,399,436,448,483,495,526,545]],"t":[[0,17,22,28,35,47,52,63,68,80,85,93,106,114,118,130,135,147,152,163,190],[506,513,519,529,535,546,552,563,569,579,585,596,602,613,619,629,635,646,652,652,663,669,680,685,696,702,713,719,729,735,746,763,769,779,785,786,794,802,810],[1087,1097,1104,1113,1129,1135,1146,1152,1163,1169,1179],[1434,1442,1445,1461,1469,1480,1486,1486,1494,1502,1513],[1765,1783,1786,1787,1794,1802,1813,1819,1830,1836,1847,1853,1863,1869,1880,1886,1889,1894],[2083,2101,2104,2106,2111,2119,2130,2136,2147,2149,2153,2161,2169,2180,2186,2199,2203,2219,2220,2228,2228,2236,2244,2253,2263,2270,2280,2286,2286,2297],[2468,2486,2489,2494,2503,2513,2519,2530,2536,2547,2553,2553,2564,2569],[2796,2813,2817,2821,2828,2836,2847,2853,2864,2869,2880,2886,2894,2903,2913,2919,2930,2936,2947,2953,2964,2969]],"version":"2.0.0"}.

    Similarly, the x-intercept is given by

    -ca=--12=12.

    Hence the x-intercept lies at12,0{"x":[[274,269,268,266,265,263,254,250,236,231,213,207,201,179,169,165,159,154,153,153,153,158,164,170,178,187,191,206,211,226,237],[366,364,363,363,361,358,357,354,352,348,345,343,342,342,341],[296,295,298,303,307,321,327,337,343,347,357,361,370,378,385,390,394],[338,337,337,335,335,335,335,336,339,342,343,350,353,361,366,370,373,374,375,374,369,363,357,350,343,338,336,332,332,334,336,344,352,362,372,378,388,393,398,412,419,426,430,433,435],[469,469,469,469,469,468,468,466,466,465],[556,556,556,556,556,556,555,554,554,554,554,557,561,565,571,574,581,589,596,600,607,615,622,628,634,636,641,643,645,645,645,645,644,642,638,633,630,618,609,600,593,587,581,580,580],[699,701,703,704,709,715,718,728,730,734,735,736,735,731,730,723,719,716,708,705,696,690,686]],"y":[[153,148,148,148,149,151,160,165,181,189,217,227,239,291,319,334,363,392,406,420,447,481,501,516,529,539,543,548,548,543,535],[168,178,181,184,193,208,213,226,231,242,249,255,257,259,263],[326,326,325,324,323,321,320,319,319,319,318,318,318,317,317,318,321],[410,410,409,407,405,401,398,394,390,386,384,379,378,378,380,384,389,393,404,408,422,433,443,454,463,472,477,486,488,494,495,498,498,498,497,497,495,494,493,491,489,488,487,487,487],[337,343,349,356,365,374,378,389,392,398],[270,269,268,267,271,279,284,305,314,338,346,366,376,385,391,393,397,398,397,396,392,385,377,365,352,344,318,309,287,280,274,262,258,253,247,244,243,245,249,254,260,266,275,277,279],[155,155,155,154,156,164,170,194,203,234,246,283,295,335,348,385,408,418,447,456,479,491,496]],"t":[[0,16,20,27,29,36,46,53,63,69,80,85,86,103,111,119,129,135,136,146,152,163,169,180,186,196,202,213,219,229,235],[599,615,621,630,636,646,652,663,669,680,686,697,702,703,711],[973,981,994,1002,1013,1019,1030,1036,1036,1047,1052,1053,1063,1069,1080,1086,1103],[1333,1338,1341,1345,1353,1363,1369,1380,1386,1397,1403,1413,1419,1430,1436,1447,1453,1465,1469,1478,1486,1497,1502,1513,1519,1533,1536,1547,1553,1564,1569,1580,1586,1595,1603,1614,1619,1620,1628,1636,1646,1653,1664,1669,1681],[1940,1947,1955,1963,1969,1980,1986,1997,2003,2011],[2232,2239,2245,2253,2270,2280,2286,2297,2303,2314,2319,2331,2336,2347,2353,2364,2370,2381,2386,2387,2397,2403,2413,2420,2430,2436,2447,2453,2464,2469,2470,2481,2486,2487,2497,2503,2514,2520,2531,2536,2547,2553,2570,2578,2586],[2848,2853,2858,2861,2870,2881,2886,2898,2903,2914,2920,2931,2936,2948,2953,2964,2971,2978,2986,2997,3003,3014,3020]],"version":"2.0.0"}.

    If we were to plot the line, then we locate the x and y-intercepts and then connect them to get the desired line.

    Gradient and Intercepts, The plot of the straight line 2x+4y-1=0, StudySmarterThe plot of the straight line 2x+4y-1=0, StudySmarter Originals

    Find the gradient of the line of equation 3x-y+1=0.

    Solution

    Comparing the given equation of line to the general form ax+by+c=0{"x":[[187,188,189,189,189,188,186,184,182,175,169,162,153,148,131,97,98,102,106,112,115,123,127,134,143,153,162,172,183,191,199,205,209,211,212,216,219,221,223,226,232,234,240,248,254,257],[279,278,277,275,275,277,279,281,288,291,295,302,306,313,319,325,327,331,332,331,329,328,323,320,312,309,306,304,300,299,299,300],[376,377,377,377,377,375,373,370,365,359,357,348,345,339,338,335,336,339,344,347,354,358,363,373,378,383,388],[443,444,450,457,462,473,484,493,497,506,512,515],[479,479,477,477,477,477,478,478,478,478],[573,571,570,568,567,565,562,562,562,562,560,559,559,559,558,558,559,559,560,563,564,567,571,574,579,586,593,596,605,607,614,616,617,618,619,619,618,617,615,614,609,606,603,598,595,588,581,575,572,567,566],[643,643,642,641,640,640,642,644,646,651,656],[688,693,695,698,700,703,704,706,706,704,703,699,697,689,686,684,681,681,680,680,681,681,682,683,683,684,685],[765,778,787,791,795,804,808,812,818,821,823,827,828,829],[799,799,797,796,795,791,790,787,786,785,785,786,787,788],[898,879,876,868,860,857,854,852,846,846,845,845,845,847,853,859,867,872,876,886,891,895],[942,944,960,964,967,971,975,979,985,988,997,999,1002],[934,941,950,954,966,970,978,982,989,995],[1054,1054,1051,1048,1046,1039,1037,1034,1033,1033,1033,1034,1035,1038,1040,1045,1047,1054,1060,1066,1069,1075,1080,1083,1085,1090,1091,1092,1091,1089,1085,1080,1074,1070,1058,1054,1050,1044],[639,638,638,639,639,640,641,642,644,645,646,648,649,650,652,653,654,657,658,662,665,669,672,679,682,689,691,694,698,700,702,706,708,709,710]],"y":[[298,295,285,280,274,269,264,261,259,253,251,251,255,258,275,369,381,388,391,389,387,381,377,369,358,344,331,316,300,290,284,283,285,287,290,317,339,346,352,358,367,369,370,370,369,367],[307,306,305,301,299,295,293,290,286,284,283,282,283,286,292,299,302,314,318,331,335,340,350,355,367,371,375,377,381,382,381,379],[294,292,289,288,287,286,286,289,293,301,305,320,325,341,346,359,364,369,374,375,377,378,378,378,377,375,373],[331,330,327,326,325,322,319,317,316,314,313,312],[286,287,290,292,301,312,331,337,354,365],[208,210,213,217,221,227,247,253,270,276,291,300,305,311,314,316,320,321,321,322,322,322,320,319,315,312,310,310,310,311,316,319,321,327,329,332,336,338,341,343,347,349,351,353,355,357,357,357,356,352,350],[297,299,303,308,312,317,326,334,338,346,349],[331,324,321,314,311,307,305,303,311,316,322,336,344,370,379,388,405,412,419,423,430,433,435,438,439,440,440],[323,320,319,318,317,316,315,314,314,313,313,313,313,314],[288,293,304,311,318,339,345,361,365,372,376,378,378,378],[270,275,278,286,295,300,305,311,329,335,346,351,356,363,371,373,373,373,372,369,366,364],[296,295,290,290,289,289,288,288,287,287,287,288,289],[328,329,329,329,327,326,325,324,324,324],[259,268,282,290,298,322,329,342,348,358,366,372,375,378,378,377,375,368,359,349,344,331,319,312,306,286,274,263,254,248,243,243,244,245,252,255,258,265],[343,343,342,342,343,343,343,343,344,344,344,345,345,345,346,347,347,349,349,350,351,351,351,349,348,344,343,341,337,336,334,330,325,323,318]],"t":[[0,6,11,18,24,36,41,41,53,57,69,74,86,91,103,142,149,157,170,174,186,191,191,203,208,219,224,236,241,253,259,269,274,286,303,307,320,324,325,336,341,341,353,358,369,374],[588,593,601,609,619,624,624,636,641,641,652,657,658,669,675,686,692,702,708,719,724,725,736,741,752,761,761,771,774,775,791,802],[913,919,926,936,941,952,958,969,975,986,991,999,1009,1019,1024,1036,1041,1053,1058,1069,1074,1075,1086,1091,1091,1103,1105],[1291,1308,1311,1313,1318,1325,1333,1341,1353,1358,1369,1371],[1557,1567,1570,1576,1583,1591,1603,1608,1620,1625],[1868,1875,1877,1882,1883,1891,1900,1909,1916,1926,1936,1941,1953,1958,1958,1970,1975,1975,1987,1991,1992,2003,2008,2009,2020,2025,2037,2041,2053,2058,2070,2075,2075,2087,2091,2092,2104,2108,2109,2120,2125,2125,2137,2141,2142,2154,2158,2170,2175,2187,2191],[2441,2452,2456,2461,2466,2470,2475,2487,2492,2503,2508],[2568,2577,2587,2591,2592,2604,2608,2620,2637,2642,2645,2654,2658,2670,2675,2675,2687,2692,2692,2704,2708,2709,2721,2725,2725,2737,2742],[3112,3129,3142,3146,3154,3159,3159,3171,3175,3175,3188,3192,3192,3200],[3370,3388,3400,3401,3408,3421,3425,3434,3442,3454,3459,3459,3467,3472],[3695,3713,3720,3727,3737,3742,3742,3754,3762,3773,3775,3776,3784,3792,3804,3810,3821,3825,3828,3838,3842,3846],[4087,4105,4108,4110,4117,4118,4125,4128,4138,4143,4154,4159,4167],[4328,4347,4359,4369,4376,4387,4392,4393,4405,4409],[4628,4647,4659,4660,4670,4676,4688,4692,4693,4705,4709,4721,4726,4738,4742,4754,4759,4776,4777,4784,4792,4804,4809,4809,4822,4826,4838,4842,4855,4859,4871,4876,4888,4893,4905,4909,4912,4922],[5729,5743,5759,5793,5805,5810,5822,5826,5838,5843,5855,5859,5860,5872,5876,5877,5889,5893,5893,5906,5910,5922,5926,5939,5943,5955,5959,5960,5972,5976,5977,5989,5993,6005,6010]],"version":"2.0.0"}, we get a=3,b=-1,c=1{"x":[[131,132,133,134,137,138,142,142,144,145,144,143,142,138,136,133,128,122,109,101,92,82,78,67,64,58,56,55,56,60,62,64,67,78,82,140,150,154,165,168,173,175,176,177,178,178,179,180,183,184,186,187],[241,249,254,261,269,276,284,288,297,300,306],[246,245,246,248,252,257,265,278,283,296,300,303,310],[354,354,360,364,368,372,375,378,380,381,382,382,375,369,368,366,365,362,361,360,360,362,365,369,374,379,382,389,390,394,395,395,395,395,394,391,389,387,383,380,378,372,368,365,359,356,350,347,345],[440,443,443,444,445,445,445,445,444,443,441,441,440,440,439],[535,538,538,539,540,541,541,541,541,541,541,540,539,538,537,535,535,534,533,532,531,531,531,530,530,530,530,530,530,531,532,533,535,541,545,551,554,564,567,577,580,582,587,588,590,591,592,592,591,590,589,585,583,578,572,566,563,554,552,550],[619,629,641,648,654,659,663,666],[599,597,598,604,607,617,621,635,639,643,647,655],[702,702,702,703,704,707,712,714,722,725],[787,788,788,788,787,785,783,781,780,778,778,777,777,777,777],[830,831,833,833,834,834,834,834,834,834,834,834,833,833,834,834],[937,937,938,938,937,937,934,932,928,925,921,914,910,898,894,891,886,884,882,883,884,887,890,893,896,903,907,912,921,926,931,942,947],[975,987,989,999,1007,1018,1024,1028,1029,1031,1032],[969,972,975,978,980,987,995,1003,1007,1011,1016,1025,1028,1032,1035],[1073,1079,1080,1081,1082,1083,1083,1082,1081,1078,1076,1075,1074,1074,1074,1074,1074,1074]],"y":[[289,288,287,285,283,281,276,274,267,263,259,254,252,248,246,244,240,237,238,243,251,262,268,288,296,318,330,340,346,350,352,353,353,352,349,302,294,292,290,291,298,302,306,315,325,336,346,350,357,360,363,366],[273,272,271,271,270,269,268,267,266,265,263],[325,328,328,328,328,326,322,317,315,310,309,308,305],[240,238,228,224,222,219,218,217,217,219,220,223,234,243,245,247,249,254,258,260,264,267,270,273,276,280,282,288,290,296,299,305,307,309,311,314,316,317,320,321,322,323,323,323,323,323,321,318,317],[340,341,342,345,347,354,357,360,367,373,379,381,383,386,387],[178,167,166,165,165,166,170,176,179,188,192,203,208,227,234,252,258,276,282,288,299,303,307,314,319,321,324,325,326,326,326,326,324,319,316,313,312,310,310,311,312,312,315,316,317,320,324,326,330,332,334,338,340,343,346,347,347,345,343,341],[265,263,262,261,260,260,260,260],[304,306,306,306,305,303,301,298,297,296,295,293],[277,278,279,279,279,279,278,278,276,276],[222,221,222,224,229,236,246,256,266,275,280,289,292,300,302],[328,329,331,333,336,342,351,357,360,363,369,372,374,379,384,386],[242,240,232,230,229,228,228,228,230,232,234,242,246,263,269,275,287,293,308,312,315,321,324,326,328,330,331,331,331,331,330,327,325],[258,254,253,251,249,248,247,247,247,247,247],[293,294,294,294,294,293,292,289,288,287,286,283,282,281,280],[202,200,201,206,209,223,228,248,257,278,293,305,316,321,331,335,340,344]],"t":[[0,6,11,19,30,35,46,51,63,67,79,84,96,101,101,113,118,130,134,143,151,163,168,180,185,196,201,213,218,226,234,235,246,251,259,304,309,318,330,335,346,351,351,363,369,382,384,396,401,402,413,415],[720,738,745,753,763,768,779,785,796,802,813],[979,997,1004,1013,1018,1029,1035,1046,1051,1063,1068,1071,1080],[1292,1310,1313,1319,1329,1336,1346,1352,1363,1372,1382,1386,1410,1426,1435,1435,1446,1451,1460,1463,1468,1480,1485,1496,1502,1513,1518,1530,1536,1546,1551,1563,1568,1568,1580,1585,1585,1597,1601,1602,1613,1618,1619,1630,1635,1635,1647,1652,1663],[1946,1965,1973,1977,1985,1997,2002,2002,2013,2018,2030,2035,2035,2043,2048],[2309,2326,2330,2335,2347,2352,2363,2372,2383,2385,2385,2393,2402,2413,2418,2430,2436,2447,2452,2452,2464,2468,2469,2481,2485,2497,2502,2502,2514,2518,2519,2527,2535,2547,2552,2564,2569,2580,2586,2597,2602,2602,2614,2618,2619,2631,2635,2647,2652,2652,2664,2669,2669,2681,2685,2697,2702,2714,2719,2723],[3063,3082,3096,3103,3114,3119,3130,3135],[3288,3304,3319,3330,3336,3347,3352,3364,3372,3372,3383,3385],[3663,3681,3688,3703,3714,3719,3731,3736,3748,3753],[3963,3982,3994,3997,4002,4014,4019,4031,4036,4048,4052,4064,4069,4081,4086],[4343,4355,4358,4364,4377,4378,4386,4394,4402,4403,4414,4419,4419,4431,4436,4448],[4727,4745,4748,4750,4756,4764,4769,4770,4782,4786,4788,4798,4803,4815,4819,4822,4832,4836,4848,4853,4853,4865,4869,4870,4882,4886,4886,4898,4903,4903,4915,4919,4920],[5168,5186,5190,5203,5215,5231,5236,5248,5253,5255,5265],[5413,5420,5431,5436,5437,5449,5453,5465,5469,5470,5482,5486,5487,5499,5501],[5668,5686,5690,5696,5703,5715,5720,5732,5736,5749,5753,5765,5770,5782,5786,5787,5799,5801]],"version":"2.0.0"}.

    The slope or the gradient of the line can be calculated via,

    m=-ab=-3-1=31=3

    Thus, the slope of the given line is 3.

    The graph of this straight line would be,

    Gradient and Intercepts, The graph of the straight line given by 3x-y+1=0, StudySmarter The graph of the straight line given by 3x-y+1=0, StudySmarter Originals

    where A and B lie at the x and y-intercepts of the line.

    Recall that the coordinates of the x-intercept are -ca,0{"x":[[241,241,241,240,238,228,222,218,205,201,189,185,182,175,172,166,165,169,171,181,201,213,227,239,247,269,276,291,298],[288,289,289,291,295,303,313,323,333,338,342,347,361,377,383,388,392,394],[506,503,500,495,490,485,482,473,470,463,459,456,454,454,456,457,464,467,470,478,486,495],[429,430,432,434,444,449,464,486,496,506,516,525,532,536,542],[508,509,510,510,511,511,511,508,506,500,495,493,480,474,471,468,463,462,461,462,464,466,469,474,476,490,495,500,506,509,512,515,516,518,518,520,526,530,532],[614,614,614,614,614,613,613,613],[719,719,720,720,720,718,717,716,715,712,712,711,711,712,715,718,722,727,729,738,741,751,754,757,762,767,769,772,772,772,771,770,765,759,751,743,740,728],[834,835,837,839,841,848,852,856,858,859,860,860,857,855,832,829]],"y":[[205,203,200,197,196,197,203,207,225,233,261,272,283,308,320,361,388,439,451,476,497,504,509,511,511,507,505,500,497],[282,282,281,280,280,279,279,279,279,279,279,279,279,277,277,277,277,277],[189,188,187,188,191,195,198,209,213,226,234,242,249,252,260,262,265,266,266,266,265,264],[334,334,333,333,332,331,329,325,323,321,319,318,316,316,315],[389,387,385,382,381,376,374,370,368,365,365,365,374,381,385,389,404,409,420,424,426,427,426,424,423,413,408,404,401,401,402,405,407,415,419,427,433,433,433],[375,376,377,386,405,414,417,421],[288,287,287,288,290,296,301,306,312,331,336,347,352,356,366,372,375,377,378,377,376,370,367,364,356,347,343,327,320,310,305,300,289,284,281,278,278,278],[240,241,242,243,245,254,263,275,288,294,316,323,346,354,420,426]],"t":[[0,11,14,24,33,58,63,79,81,88,99,104,105,115,121,132,141,162,165,177,188,198,205,214,221,233,243,246,247],[662,666,681,690,698,704,714,721,731,738,738,750,759,775,780,788,799,804],[1155,1162,1167,1172,1181,1188,1196,1205,1213,1221,1231,1238,1248,1256,1269,1271,1283,1288,1291,1299,1305,1313],[1608,1617,1622,1633,1638,1648,1655,1674,1684,1688,1700,1705,1715,1722,1731],[1975,1981,1988,1998,2005,2015,2022,2031,2038,2047,2063,2065,2085,2088,2089,2097,2105,2115,2122,2132,2138,2149,2155,2165,2173,2193,2201,2205,2217,2222,2232,2239,2248,2255,2265,2273,2293,2301,2305],[2541,2547,2563,2575,2595,2597,2607,2610],[2912,2923,2932,2955,2956,2966,2972,2972,2986,2989,3001,3005,3006,3016,3022,3032,3039,3049,3055,3066,3072,3085,3089,3089,3101,3105,3116,3122,3133,3139,3139,3149,3155,3164,3172,3181,3189,3201],[3421,3425,3434,3439,3449,3456,3466,3472,3483,3491,3501,3506,3516,3522,3557,3564]],"version":"2.0.0"} and for the y-intercept 0,-cb{"x":[[259,255,253,246,241,235,228,220,213,205,197,193,183,180,172,169,168,168,168,174,176,187,191,206,217,229,241,248,267,273],[439,437,437,434,434,432,432,431,429,428,427,426,426,426,427,433,436,444,448,458,467,477,482,488,497,506,515,517,524,525,525,523,518,511,507,491,481,471,461,453,450],[603,604,604,604,603,603,603,603,603,603],[664,662,665,669,675,679,683,687,703,708,723,730,736,738],[846,849,849,848,845,843,834,831,819,811,804,797,790,785,783,782,787,789,797,804,812,820,825,839,843,855],[753,758,764,772,783,795,800,817,828,837,846,854,856,858,861],[809,811,812,813,813,813,810,809,805,803,799,797,796,793,792,792,796,798,800,805,811,815,825,828,837,841,844,846,846,846,845,841,839,836,828,825,818,815,809,804,802],[931,939,952,955,959,962,972,975,982,984,989,991,992,993,993,993,990,988,985,982,975,967,958,954]],"y":[[211,209,209,211,216,225,236,249,265,283,303,313,346,356,391,415,440,463,474,504,512,532,538,549,553,554,554,554,550,548],[321,320,319,319,320,327,332,337,350,357,364,379,385,391,397,412,415,418,418,417,414,407,403,399,389,378,364,359,339,333,316,307,300,297,296,296,299,304,310,316,319],[394,395,398,403,405,409,415,419,427,430],[301,300,301,302,303,303,303,303,302,301,298,296,295,295],[254,244,243,240,238,237,236,237,240,245,251,259,268,278,287,295,306,309,313,314,315,315,314,312,311,309],[386,386,386,386,385,384,383,380,379,377,376,376,376,376,376],[429,432,433,437,443,446,460,465,479,484,497,500,503,508,512,514,512,511,509,505,501,499,496,496,496,497,499,501,503,505,506,509,510,511,514,515,515,515,515,512,511],[227,217,226,230,234,239,258,265,292,301,331,354,375,385,397,408,431,442,454,466,492,515,538,549]],"t":[[0,10,14,22,30,39,47,55,64,74,80,91,97,107,114,125,130,141,147,158,164,175,180,191,197,207,215,226,230,242],[4488,4492,4501,4509,4515,4525,4532,4532,4542,4549,4549,4559,4565,4566,4576,4582,4592,4599,4599,4612,4615,4627,4632,4635,4642,4649,4660,4665,4676,4682,4696,4699,4711,4715,4724,4732,4742,4749,4761,4767,4776],[5040,5044,5050,5057,5066,5068,5076,5082,5093,5099],[5414,5428,5441,5449,5459,5466,5466,5476,5482,5493,5499,5509,5516,5523],[5772,5782,5791,5799,5809,5816,5826,5832,5842,5849,5859,5866,5876,5882,5895,5899,5919,5924,5932,5943,5949,5959,5966,5976,5983,5993],[6252,6261,6267,6276,6283,6293,6299,6310,6316,6326,6333,6343,6349,6350,6360],[6544,6557,6560,6566,6577,6583,6593,6600,6610,6616,6627,6633,6633,6643,6649,6662,6683,6692,6697,6700,6710,6716,6727,6733,6747,6749,6760,6766,6776,6783,6795,6799,6800,6812,6816,6827,6833,6834,6845,6849,6857],[7164,7165,7168,7175,7175,7183,7194,7200,7210,7216,7227,7233,7243,7250,7250,7260,7266,7267,7277,7280,7283,7294,7300,7308]],"version":"2.0.0"}.

    Using this, the x-intercept of our line 3x-y+1=0{"x":[[239,240,240,241,242,249,251,259,264,268,269,272,273,273,273,272,269,264,259,254,250,248,244,244,244,245,250,254,259,264,267,272,274,276,281,283,283,282,279,275,272,262,258,245,240,236],[332,331,330,329,331,332,339,344,351,354,363,365,368,370,372,373,373,372,367,362,356,347,343,341],[394,393,393,392,390,387,385,381,379,377,374,373,372,370,370,372,375,377,385,389,401,406],[483,484,487,488,497,500,504,511,519,526,530,540,545,548,552,554],[606,609,610,612,613,614,615,616,616,618,621,627,629,634,638,643,647,649,656,658,664,666,668,669,669,669,669,668,667,666,666,665,664,663,663,662,661,659,658,654,653,649,647,644,639,633,630],[722,724,726,731,737,745,752,759,763,772,774,779],[767,767,767,766,766,763,761,757,753,750,748,748,748,749,749],[847,847,848,849,850,850,849,848,847,846,846],[847,848,848,849,850,852,854],[938,938,939,941,947,950,961,965,976,984,991,997,1002,1005,1006],[939,937,937,938,943,949,957,962,972,977,985,989,1000,1003,1006,1010,1012,1015],[1073,1075,1075,1074,1072,1068,1065,1061,1059,1059,1058,1059,1063,1065,1067,1072,1077,1080,1083,1088,1093,1098,1102,1104,1105,1105,1101,1084,1077]],"y":[[241,239,237,234,232,227,225,221,221,221,222,225,228,230,232,241,247,254,261,267,273,277,283,285,290,291,295,297,300,303,305,309,311,313,320,325,329,335,340,345,348,355,358,363,364,365],[278,277,275,273,271,270,269,268,268,268,270,272,273,277,281,283,290,293,303,310,316,325,329,331],[276,271,270,270,271,275,278,284,288,292,300,304,308,316,325,330,334,335,339,340,341,341],[299,300,299,298,296,296,295,295,294,293,293,291,290,290,289,289],[263,263,263,266,270,275,281,287,289,293,298,299,299,297,293,289,283,280,269,265,255,253,248,248,249,250,252,260,264,272,276,286,296,306,318,327,332,346,350,360,363,368,369,370,371,369,367],[271,271,271,271,271,271,271,271,271,270,270,270],[242,241,240,242,244,255,266,278,290,299,307,311,318,320,322],[241,242,244,250,260,266,282,287,302,311,315],[329,332,335,337,339,342,343],[268,267,266,266,265,265,264,264,264,264,264,264,264,265,265],[297,298,299,299,299,299,299,298,297,297,296,295,294,294,294,293,293,293],[257,254,256,264,272,282,294,305,314,318,328,330,334,335,335,335,333,331,329,323,316,307,297,285,273,267,252,238,238]],"t":[[0,7,12,18,25,36,42,53,59,69,75,86,92,92,103,109,120,125,136,142,153,159,170,178,186,192,203,209,219,225,236,242,243,253,259,267,275,286,292,303,309,320,325,336,342,346],[578,584,592,602,619,625,636,642,652,659,669,675,676,686,692,702,709,719,726,736,744,761,767,768],[935,946,951,959,969,976,986,992,993,1003,1009,1009,1019,1026,1036,1042,1052,1059,1071,1076,1086,1092],[1331,1342,1351,1359,1369,1376,1376,1386,1392,1403,1409,1420,1426,1436,1442,1443],[1658,1668,1673,1686,1692,1703,1709,1719,1726,1737,1744,1760,1768,1776,1786,1793,1803,1809,1819,1826,1836,1843,1851,1859,1876,1877,1886,1893,1903,1909,1910,1920,1926,1936,1943,1953,1959,1970,1976,1986,1993,2003,2009,2010,2020,2026,2036],[2343,2351,2360,2370,2376,2387,2393,2403,2409,2420,2426,2434],[2611,2623,2629,2643,2652,2660,2670,2676,2687,2693,2703,2710,2720,2726,2731],[2963,2977,2985,2993,3004,3010,3020,3026,3039,3043,3051],[3067,3070,3076,3079,3087,3093,3093],[3865,3877,3886,3893,3904,3910,3921,3927,3937,3943,3954,3961,3975,3977,3988],[4114,4126,4133,4138,4143,4154,4160,4171,4177,4177,4188,4193,4206,4210,4212,4221,4227,4235],[4428,4434,4452,4460,4470,4477,4487,4494,4504,4510,4521,4527,4540,4544,4544,4554,4560,4571,4573,4577,4588,4594,4604,4610,4619,4627,4637,4664,4669]],"version":"2.0.0"} is -ca,0=-13,0{"x":[[149,146,143,139,137,129,125,122,113,103,93,84,79,70,68,64,64,68,73,79,86,95,100,105,114,124,132,140,144],[139,138,139,144,146,155,163,170,177,180,188,190,195],[308,302,298,297,295,290,285,282,274,272,266,264,263,263,266,269,272,277,279,287,291,300,303],[260,261,261,267,274,283,292,301,305,308,316,319,322,324,329,330,331],[312,314,315,316,316,317,317,317,317,315,314,309,307,301,297,292,288,284,281,280,279,279,282,286,290,295,299,302,308,311,316,321,326,328,331,333,336,337,338,339,339,340,340,340,341,342,343,345],[391,390,390,390,391,391,391,391],[450,450,450,450,450,450,450,450,451,451,453,455,457,460,464,466,470,472,476,479,480,480,479,476,474,471,469,466],[509,511,512,513,515,516,517,518,519,523,525,527,528,529,529,528,526,523,518,513,511,502,499,491,487,485],[588,586,589,593,598,605,614,622,626,634,640,644,646,648,648],[594,595,597,601,607,615,622,625,635,640,644,646,650],[780,779,778,775,772,768,765,755,752,741,733,726,722,720,714,712,708,708,709,711,713,717,723,730,737,746],[767,767,766,768,769,772,780,784,792,796,799,810,816,820],[890,892,892,893,893,892,890,890,888,887,886,885,885,885,887,888,889],[843,848,849,851,854,859,862,872,876,886,889,892,894],[863,861,861,862,866,869,873,876,880,883,884,887,887,885,882,879,875,873,867,865,860,858,858,858,860,863,866,869,870,873,874,875,876,876,875,873,871,869,863,861,854],[949,950,951,951,951,950,948,947,944,943,941,941,940,941,941],[1013,1013,1013,1013,1013,1011,1010,1009,1008,1007,1007,1007,1009,1012,1013,1018,1020,1026,1028,1034,1038,1042,1044,1045,1046,1047,1046,1044,1042,1040,1031,1026,1021,1016],[1086,1086,1086,1087,1088,1090,1093,1095,1098,1098,1099,1099,1098,1095,1094,1090,1086,1082,1078,1075,1070,1067]],"y":[[202,197,196,195,195,198,200,203,210,221,236,254,265,298,310,346,357,391,410,425,437,445,449,452,457,460,461,460,459],[311,311,311,311,311,309,308,307,306,305,304,304,303],[216,213,212,212,212,213,216,217,226,230,242,246,257,260,268,272,275,277,278,279,279,277,276],[322,322,323,323,322,320,319,317,316,315,314,313,313,313,312,312,313],[383,383,383,382,381,379,377,375,372,368,366,362,361,360,361,365,370,377,385,389,401,404,411,412,412,411,408,406,402,399,393,388,383,381,378,377,377,378,381,383,389,395,404,406,409,413,416,418],[349,355,359,364,369,375,380,382],[282,283,284,288,291,304,309,324,331,335,340,343,345,347,347,347,342,340,330,323,315,307,299,292,290,285,284,282],[229,226,226,227,231,236,239,243,248,263,274,285,291,302,308,321,334,347,362,374,380,396,401,412,417,419],[277,277,277,277,277,276,275,274,274,274,274,274,275,277,278],[309,309,309,309,308,308,307,307,307,307,307,307,307],[221,221,221,221,222,226,228,240,245,264,279,295,303,311,328,337,362,377,383,395,400,408,415,420,422,423],[305,306,306,306,306,306,305,305,303,302,302,300,298,297],[228,227,226,225,227,229,233,236,242,246,257,261,271,274,285,289,290],[327,328,328,328,328,327,326,325,324,323,323,323,323],[371,373,372,372,370,369,368,367,367,368,369,372,374,379,383,386,390,392,397,398,403,405,407,409,411,413,416,418,419,421,422,423,425,426,427,428,429,429,429,429,427],[347,347,348,350,353,358,365,369,379,382,390,392,397,398,399],[303,304,306,308,314,326,330,335,343,347,357,362,365,367,368,369,369,365,363,354,347,339,334,330,321,313,309,303,301,299,294,292,290,289],[256,254,253,254,258,264,272,284,295,301,308,321,327,347,354,376,390,404,414,420,429,434]],"t":[[0,11,16,24,33,40,50,52,57,67,74,84,90,103,107,119,124,137,140,149,157,167,174,174,184,190,201,207,215],[807,815,841,851,857,868,874,887,891,903,907,921,924],[1317,1328,1333,1341,1341,1351,1357,1367,1374,1384,1391,1399,1408,1417,1424,1438,1441,1454,1457,1468,1474,1484,1490],[1718,1733,1741,1751,1758,1768,1774,1785,1791,1791,1802,1808,1808,1818,1824,1834,1837],[2053,2060,2066,2074,2075,2085,2091,2091,2102,2108,2117,2124,2135,2141,2154,2158,2168,2174,2185,2191,2205,2208,2217,2224,2235,2241,2251,2258,2268,2271,2275,2288,2291,2303,2308,2308,2321,2326,2335,2341,2351,2358,2370,2375,2377,2385,2391,2404],[2643,2645,2657,2658,2668,2675,2685,2691],[2962,2983,2991,3002,3008,3018,3025,3036,3041,3052,3059,3060,3072,3075,3087,3091,3100,3108,3118,3125,3138,3141,3154,3158,3169,3175,3175,3183],[3398,3408,3411,3416,3425,3435,3442,3442,3452,3458,3469,3475,3486,3492,3492,3502,3508,3519,3525,3535,3542,3554,3561,3569,3575,3583],[3985,3996,4017,4025,4035,4042,4052,4059,4069,4075,4084,4092,4092,4102,4108],[4301,4325,4326,4336,4342,4355,4359,4367,4375,4386,4392,4403,4409],[4699,4703,4710,4719,4726,4736,4742,4753,4759,4769,4775,4786,4792,4792,4803,4809,4820,4825,4836,4842,4843,4853,4859,4869,4875,4886],[5186,5194,5203,5219,5226,5236,5242,5253,5259,5259,5270,5276,5286,5292],[5541,5544,5551,5559,5576,5586,5592,5593,5603,5609,5620,5626,5636,5644,5660,5668,5676],[5906,5917,5920,5924,5928,5937,5943,5953,5959,5970,5976,5976,5984],[6223,6230,6251,6259,6270,6276,6287,6293,6303,6309,6320,6326,6337,6343,6353,6359,6370,6376,6387,6393,6403,6409,6422,6426,6437,6443,6453,6459,6472,6476,6477,6487,6493,6493,6504,6509,6519,6526,6537,6543,6554],[6767,6772,6785,6793,6803,6810,6820,6826,6837,6843,6854,6860,6871,6876,6884],[7156,7176,7187,7193,7204,7210,7219,7225,7228,7237,7243,7255,7260,7271,7276,7287,7293,7308,7310,7321,7326,7337,7343,7343,7354,7360,7371,7376,7377,7390,7393,7406,7410,7421],[7659,7660,7663,7677,7687,7693,7705,7710,7721,7727,7729,7738,7743,7754,7760,7771,7777,7788,7793,7804,7810,7810]],"version":"2.0.0"} and the y-intercept is 0,-cb=0,1{"x":[[135,125,123,120,112,110,101,95,88,85,82,79,74,71,69,63,60,59,59,60,63,64,71,74,83,87,99,106,110,118],[166,163,160,156,154,153,152,152,154,155,160,162,169,174,180,183,187,193,195,200,204,206,207,208,208,206,203,197,191,184,180,169],[275,275,275,275,275,275,275,275,275,275,275,275],[326,325,326,328,333,339,347,357,361,372,376,384,386,389],[500,493,489,484,478,473,467,465,457,455,450,450,451,452,454,458,464,467,478,482,494,498],[424,440,444,457,472,485,489,499,504,508],[464,464,464,464,463,463,460,459,456,454,453,452,451,451,451,451,452,454,456,458,459,462,466,471,473,480,482,489,492,495,497,497,497,496,492,488,483,478,473,469,467,462],[577,578,580,583,587,590,594,599,600,601,601,601,599,597,593,590,582,579,571,568,559,554,549,545],[641,642,646,651,658,665,671,678,681,684,689,691],[640,641,642,645,650,653,657,661,672,676,680,686,692,698,702,706],[851,851,851,850,847,845,839,831,822,818,806,801,791,788,782,779,779,780,782,784,788,792,802,810,819,828,832],[901,900,899,898,897,894,891,889,887,887,886,886,889,891,897,902,907,913,919,922,924,926,930,930,930,928,926,917,909,904,891],[978,981,982,983,983,984,984],[1062,1063,1063,1063,1063,1062,1061,1060,1060,1057,1057,1055,1055,1054,1054],[1078,1085,1087,1091,1094,1097,1099,1102,1105,1107,1109,1109,1108,1104,1101,1095]],"y":[[191,197,199,202,214,219,237,250,266,275,285,294,315,327,338,374,399,424,447,469,490,498,522,529,544,548,557,559,560,559],[328,331,335,346,357,369,380,391,400,404,412,414,416,415,411,408,405,396,392,381,369,363,357,351,340,331,323,318,316,315,316,322],[387,388,389,390,391,396,399,408,412,415,422,424],[320,320,321,321,321,321,321,321,320,319,318,317,316,315],[233,230,230,231,234,238,243,247,260,264,278,286,293,296,299,303,306,307,309,309,309,309],[372,368,368,365,362,360,359,358,358,358],[411,412,413,414,417,419,428,433,445,453,460,463,472,476,477,480,480,479,477,475,473,470,465,462,460,457,456,456,458,460,463,466,468,472,478,482,486,489,491,492,492,487],[246,247,248,251,258,263,274,294,301,326,334,360,379,396,414,422,448,456,478,484,501,510,516,519],[344,343,343,343,343,343,343,343,344,344,344,344],[387,388,388,388,388,387,387,386,384,384,383,382,382,381,381,381],[272,271,269,268,268,269,274,283,295,302,327,337,364,374,404,423,441,457,465,471,483,489,501,506,509,511,511],[356,352,352,352,353,361,369,378,388,393,407,410,419,420,423,423,422,418,412,409,406,402,389,385,378,374,371,366,365,365,368],[417,422,426,431,434,439,446],[352,354,355,356,358,363,367,371,376,389,393,405,412,417,419],[306,305,307,313,317,326,332,343,356,368,382,397,404,427,436,450]],"t":[[0,7,12,19,30,35,46,52,62,68,68,79,85,85,96,102,113,118,129,135,146,152,162,168,179,185,196,202,212,218],[664,673,676,685,695,702,712,718,729,735,745,752,762,768,777,785,785,795,802,812,818,819,829,831,836,847,852,862,868,879,885,897],[1186,1186,1189,1191,1194,1212,1218,1229,1235,1235,1245,1252],[1553,1568,1585,1595,1602,1612,1618,1629,1635,1645,1652,1662,1668,1677],[2046,2057,2063,2069,2079,2085,2093,2102,2112,2119,2129,2135,2145,2152,2152,2162,2168,2180,2185,2196,2202,2213],[2507,2519,2529,2535,2552,2560,2569,2579,2585,2595],[2812,2824,2827,2838,2844,2852,2862,2869,2879,2885,2896,2902,2916,2919,2930,2935,2948,2953,2964,2968,2971,2979,2986,2995,3002,3012,3019,3029,3035,3044,3052,3062,3069,3079,3086,3097,3102,3115,3119,3131,3135,3147],[3463,3469,3479,3485,3496,3502,3512,3519,3529,3538,3544,3552,3562,3569,3579,3585,3596,3602,3612,3619,3629,3635,3646,3652],[3947,3969,3985,3995,4002,4014,4019,4031,4035,4038,4048,4052],[4218,4223,4228,4235,4246,4252,4252,4263,4269,4279,4281,4286,4296,4302,4313,4319],[4576,4579,4586,4596,4602,4603,4613,4619,4629,4635,4646,4652,4663,4669,4679,4685,4696,4702,4710,4713,4719,4729,4735,4746,4752,4763,4769],[5000,5011,5014,5019,5029,5036,5046,5052,5063,5069,5080,5086,5096,5102,5113,5119,5128,5135,5146,5152,5153,5161,5169,5179,5186,5186,5194,5202,5213,5219,5229],[5495,5507,5512,5519,5529,5538,5543],[5865,5877,5880,5886,5886,5896,5902,5903,5913,5919,5930,5936,5946,5952,5960],[6179,6191,6196,6203,6213,6219,6220,6230,6236,6247,6252,6263,6269,6278,6286,6294]],"version":"2.0.0"}.

    Gradient and Intercepts applications: Slope-Intercept form of a line

    Just as a straight line can be generally expressed by the form ax+by+c=0{"x":[[211,211,156,142,133,133,135,136,142,148,150,161,164,172,181,193,201,209,215,218,222,224,228,231,233,236,239,242,245,246,249],[267,268,270,274,279,285,292,298,300,305,310,314,315,316,315,313,308,306,299,295,292,289,287],[339,339,340,340,340,339,337,336,335,331,330,326,322,319,318,316,316,317,318,323,327,332,335,339,342,352,356],[406,409,414,420,429,433,437,444,452,456,466,472,477,481,483,485,488,489,489],[448,448,448,448,448,450,451,452,454,454,454,453,453,450,449,448,447,447,447,449,450],[538,537,537,536,535,533,530,528,527,525,524,522,520,519,518,518,518,517,517,517,518,520,522,525,529,532,534,539,544,549,551,555,557,559,561,562,562,562,562,559,557,554,552,550,546,542,536,535,533,532,532,532],[598,597,597,596,596,597,597,598,600,602,605,609,613,618,622,627,632,639,642,647,649,652,654,655,656,656,654,652,651,649,648,645,642,639,638,637,635,635,634,633,633,633,634,634,637,637],[692,694,696,697,701,705,712,714,724,727,733,740,746,751,756,760,763,764,765],[749,749,749,749,749,749,748,748,748,748,747,747,747,747,747,747,747,748,749,750],[857,859,859,859,857,854,850,847,845,841,834,830,827,816,812,805,803,802,801,800,801,802,806,809,812,818,822,831,841,851,863,873,879,889],[928,930,935,940,944,954,957,960,966,971,974,979,984,985,986,986,985],[925,937,941,944,956,960,971,974,983,986,988,992],[1048,1048,1048,1046,1045,1041,1040,1037,1035,1033,1032,1032,1032,1032,1033,1034,1040,1045,1050,1055,1061,1068,1071,1080,1083,1092,1094,1099,1101,1102,1102,1102,1100,1096,1094,1089,1087,1085,1079,1070,1066,1062,1060,1059,1057,1056]],"y":[[337,332,325,353,392,399,403,405,407,406,405,398,395,388,379,366,358,352,350,350,352,354,361,368,373,384,391,396,400,402,407],[350,348,344,339,335,332,331,331,332,335,339,348,356,363,371,376,389,394,408,415,421,426,429],[356,355,353,351,350,350,351,352,353,356,358,364,370,378,383,395,399,410,412,419,422,424,424,424,424,422,421],[381,381,381,381,380,379,379,377,376,376,375,374,374,374,374,373,373,372,371],[352,351,348,347,346,346,347,350,355,361,367,379,383,395,399,405,411,416,419,421,421],[275,276,277,280,287,299,312,324,329,340,346,355,364,369,375,378,381,386,388,389,390,390,388,386,382,380,378,375,372,372,372,372,373,375,377,380,382,383,387,393,395,399,402,403,407,409,411,411,410,407,404,402],[349,353,355,360,365,371,376,380,383,386,387,388,388,386,382,378,373,364,360,350,347,341,338,337,338,343,349,358,363,373,379,392,404,417,423,430,440,445,450,460,472,475,481,482,483,482],[351,351,351,351,351,352,352,352,352,352,351,350,349,348,346,345,344,343,343],[317,318,322,324,328,337,341,346,354,359,362,366,372,375,378,382,383,387,389,390],[321,316,314,313,311,311,310,310,311,313,317,321,324,338,344,361,367,373,378,388,392,395,401,404,407,410,412,414,415,415,413,409,407,402],[337,337,337,336,335,334,334,333,332,332,332,332,333,334,335,336,337],[377,376,376,376,373,372,369,368,366,366,365,364],[304,307,309,318,323,339,344,362,375,387,393,399,405,413,417,420,426,429,431,431,429,425,423,413,409,393,388,370,357,351,339,333,323,313,309,301,298,295,291,287,288,291,294,296,303,306]],"t":[[0,18,42,60,80,90,94,103,114,122,127,135,147,152,160,173,177,188,194,205,210,211,225,228,235,247,259,261,273,277,285],[607,612,619,627,637,643,659,661,673,677,688,694,705,711,721,727,738,744,759,761,772,777,788],[924,929,940,944,962,973,977,978,988,994,994,1005,1011,1022,1027,1038,1044,1059,1061,1073,1077,1088,1094,1094,1102,1110,1121],[1337,1344,1353,1361,1373,1377,1377,1389,1394,1405,1410,1422,1427,1438,1445,1448,1454,1462,1473],[1583,1588,1597,1602,1611,1619,1627,1639,1644,1656,1664,1673,1677,1689,1694,1705,1711,1722,1727,1739,1741],[1933,1953,1963,1973,1977,1989,1994,2002,2006,2012,2022,2027,2036,2039,2045,2052,2056,2062,2076,2078,2086,2094,2105,2111,2119,2123,2127,2139,2144,2155,2158,2163,2176,2178,2190,2194,2195,2205,2211,2224,2229,2236,2241,2244,2255,2263,2276,2278,2290,2294,2295,2306],[2513,2522,2528,2539,2545,2555,2564,2576,2578,2590,2594,2605,2611,2622,2628,2639,2644,2655,2665,2676,2679,2690,2694,2695,2711,2722,2728,2739,2741,2746,2756,2761,2776,2778,2779,2790,2794,2795,2806,2811,2822,2829,2839,2846,2856,2861],[3089,3095,3107,3112,3123,3128,3139,3146,3156,3161,3176,3178,3190,3194,3206,3211,3222,3230,3241],[3396,3428,3438,3444,3445,3456,3461,3462,3474,3478,3478,3490,3495,3495,3506,3511,3512,3523,3529,3539],[3789,3806,3813,3820,3828,3839,3846,3853,3853,3862,3870,3870,3879,3889,3895,3906,3911,3912,3923,3928,3929,3940,3945,3945,3957,3961,3962,3973,3979,3989,3995,4006,4012,4023],[4317,4322,4329,4337,4345,4356,4362,4362,4373,4378,4390,4395,4407,4412,4423,4429,4440],[4585,4602,4606,4612,4623,4629,4640,4646,4657,4662,4662,4670],[4888,4907,4913,4923,4929,4940,4945,4957,4962,4973,4978,4979,4990,4995,4996,5007,5012,5023,5029,5040,5045,5057,5063,5073,5079,5090,5095,5107,5112,5123,5129,5129,5141,5146,5155,5162,5165,5174,5179,5190,5197,5207,5213,5215,5224,5231]],"version":"2.0.0"}, we can also derive a general form determined by the slope and the intercept of the line.

    If we rearrange the given equation to get y on one side of the equation, we have

    by=-ax-c{"x":[[103,102,101,101,101,103,105,106,110,116,122,126,132,138,145,151,155,165,169,179,185,190,195,197,201,203,205,208,209,210,211,211,212,212,212,212,209,209,207,207,206,206,206,206,206,206,206,205,205,205,204,204,204,204,204],[17,15,14,14,14,14,14,14,13,12,10,9,8,8,8,8,8,8,8,9,9,10,12,15,19,21,28,31,41,43,50,53,54,56,57,57,57,57,56,55,52,50,47,43,35,33,30,25,22,17],[300,295,294,293,291,290,289,290,291,293,298,301,309,312,323,330,337,341],[293,295,297,302,308,316,324,328,332,335,347,350,357,365,369],[448,450,451,457,459,467,470,472,477,482,484,488,494,497,498,500,502,504],[611,611,611,611,611,610,609,608,604,599,594,587,584,577,574,568,563,559,556,556,556,556,558,559,564,566,568,573,579,587,590,594,601,605,610,614,621,624,626,632,634,638,640,644,645,648,649,652],[676,677,681,687,690,699,702,704,709,711,713,716,718,719,719,720,720,719,717,714,712,706,704,697],[736,734,733,732,731,728,727,725,725,725,725,725,726,727,731,735,739,742,745,752,760,764,772],[829,828,829,832,835,846,850,854,863,868,871,878,881,884,889,890,892,893,895,896],[1030,1032,1036,1039,1041,1042,1043,1043,1041,1040,1032,1029,1016,1011,996,985,980,972,969,966,962,961,960,961,963,965,972,977,987,999,1013,1021,1028,1043,1051,1073,1079]],"y":[[266,267,268,269,279,290,303,309,321,335,342,347,350,351,351,349,347,339,336,323,313,303,291,287,278,274,271,265,264,264,265,267,275,279,299,307,336,347,381,390,399,408,424,431,442,453,458,466,470,477,482,484,483,482,476],[223,221,221,222,226,234,247,255,281,290,317,325,346,357,365,368,371,375,376,376,375,371,362,354,346,342,333,331,332,334,343,347,350,357,361,365,369,375,378,381,387,389,391,395,399,399,399,398,397,392],[285,285,285,285,285,285,284,283,283,283,281,280,279,279,278,278,277,277],[325,325,324,323,322,320,319,318,318,317,316,316,315,314,314],[304,304,304,303,303,302,302,302,302,303,303,303,303,303,303,303,303,304],[274,272,266,263,257,252,251,249,247,247,247,250,253,259,263,272,282,293,303,311,315,318,323,325,327,327,327,325,321,316,313,309,302,299,295,293,290,290,291,294,296,302,305,316,319,328,329,332],[248,246,240,237,236,235,235,237,240,243,246,254,258,263,267,276,281,289,295,301,305,312,314,320],[250,248,247,248,251,260,264,280,286,293,302,307,318,320,326,329,331,331,331,330,328,326,321],[300,299,298,297,296,294,294,293,293,293,293,293,293,293,293,293,293,293,293,293],[274,273,270,266,262,261,258,255,252,251,249,249,250,252,261,271,277,290,297,304,319,325,331,342,346,349,355,357,361,363,363,363,362,358,357,351,349]],"t":[[0,8,14,22,30,41,46,58,63,75,80,91,97,108,113,125,131,141,146,158,165,177,180,192,196,197,208,213,225,230,241,246,258,263,275,280,291,296,308,313,314,325,330,331,342,347,358,363,365,375,380,392,408,413,425],[3778,3799,3802,3814,3826,3832,3842,3848,3859,3865,3876,3882,3892,3898,3909,3914,3915,3926,3932,3948,3959,3965,3973,3981,3992,3998,4009,4015,4026,4031,4042,4048,4048,4059,4065,4065,4076,4081,4082,4093,4098,4099,4109,4115,4126,4131,4134,4143,4149,4159],[4659,4677,4685,4694,4698,4710,4715,4740,4740,4748,4759,4765,4776,4781,4793,4798,4809,4815],[5051,5069,5075,5083,5093,5098,5110,5115,5115,5123,5132,5143,5148,5160,5165],[6225,6257,6265,6277,6282,6293,6299,6299,6310,6316,6327,6332,6343,6349,6360,6362,6366,6377],[6699,6704,6711,6716,6719,6727,6732,6733,6744,6749,6757,6766,6777,6782,6783,6794,6799,6810,6816,6827,6832,6834,6844,6849,6860,6866,6866,6877,6882,6894,6899,6899,6911,6916,6916,6927,6932,6933,6944,6949,6950,6961,6966,6974,6982,6994,7000,7010],[7296,7303,7306,7308,7316,7326,7332,7333,7344,7349,7350,7361,7366,7366,7378,7382,7383,7394,7399,7411,7417,7427,7433,7444],[7660,7677,7684,7694,7699,7711,7717,7727,7733,7735,7744,7749,7761,7766,7777,7783,7794,7799,7800,7811,7816,7828,7833],[9795,9812,9819,9828,9834,9845,9850,9852,9862,9867,9870,9879,9883,9884,9892,9900,9901,9912,9917,9917],[10222,10228,10236,10245,10250,10259,10263,10267,10278,10284,10296,10301,10312,10317,10329,10334,10345,10350,10351,10362,10367,10368,10379,10384,10384,10396,10400,10401,10412,10417,10429,10434,10435,10446,10451,10462,10468]],"version":"2.0.0"}

    y=-abx-cb{"x":[[67,67,67,67,68,68,68,69,69,69,72,75,77,80,86,91,96,101,107,109,116,118,125,127,131,132,133,135,136,136,136,136,136,136,135,134,132,131,128,127,125,122,119,118,115,115,113,113,113,113,113],[210,212,221,225,231,235,244,247,252,257,262,263,264],[208,217,219,225,229,238,242,252,255,264,267,270,272,275],[368,368,369,370,376,379,389,396,404,411,415,424,428,432,435,436],[525,526,526,527,529,530,533,535,537,538,540,542,543,544,544,544,544,543,540,537,533,528,522,518,508,505,495,493,486,485,483,483,483,485,486,488,492,497,503,506,514,521,529,532,543,545,552,554,557,558,558,558,557,556,556,555,555,556,557,560,562],[473,480,485,492,501,506,515,520,525,535,540,545,554,559,567,575,581,586,588],[547,548,548,548,548,549,549,549,549,549,549,549,549,549,549,549,549,549,550,551,552,555,556,560,562,567,568,571,574,576,579,582,583,585,585,585,584,582,580,574,572,565,563,557,555,551,550,550,549],[638,637,637,636,636,636,637,639,642,646,649,658,661,669,671,676,678,679,680,680,679,677,675,673,669,667,662,658,655,650,648,646],[712,712,711,709,708,704,699,697,691,689,684,681,680,680,681,681,685,687,692,697,707,710,714,718,726],[780,790,793,796,801,804,817,822,835,842,849,854,857,860],[982,983,984,984,984,984,981,977,972,966,958,954,941,937,927,925,923,920,920,920,922,926,929,932,936,943,947,952,962,968,979,984,990,1001,1006,1019],[917.5,917.5,917.5,918.5,923.5,925.5,936.5,941.5,956.5,961.5,973.5,984.5,996.5,1008.5,1013.5,1018.5,1028.5,1034.5,1041.5,1047.5,1052.5,1054.5,1055.5],[990.5,990.5,990.5,989.5,988.5,986.5,983.5,982.5,980.5,979.5,979.5,979.5,979.5,979.5,979.5,980.5,980.5,981.5,982.5,983.5,985.5,986.5,991.5,992.5,994.5,997.5,1002.5,1004.5,1007.5,1013.5,1015.5,1017.5,1017.5,1018.5,1018.5,1017.5,1015.5,1012.5,1006.5,1003.5,1001.5,999.5,994.5,992.5,987.5,984.5,983.5,983.5]],"y":[[253,254,255,257,259,271,276,291,298,306,317,321,322,325,326,326,324,321,316,314,305,302,292,289,281,280,278,276,276,277,278,280,284,287,295,303,313,324,337,344,358,374,394,401,420,426,441,450,454,459,463],[296,296,296,296,297,297,298,298,299,300,302,302,303],[336,336,336,335,335,333,333,332,332,331,331,331,330,330],[297,298,298,298,297,296,294,292,291,289,289,287,286,286,285,285],[275,275,274,274,271,270,266,262,257,255,250,245,240,234,232,229,226,222,218,216,215,215,218,220,228,232,244,248,260,264,274,280,282,286,287,288,289,290,289,288,285,281,277,274,268,266,263,263,263,265,268,276,278,281,283,287,292,294,296,297,297],[358,358,357,356,354,354,352,351,351,349,349,349,348,348,347,347,347,347,347],[406,406,407,410,411,418,424,427,433,435,438,444,445,448,449,450,451,452,452,452,451,449,448,444,443,441,441,440,440,440,441,442,443,446,450,453,457,462,464,469,471,475,476,476,475,470,467,464,461],[310,310,309,308,305,303,301,298,296,296,295,297,298,303,305,312,317,320,327,330,334,341,345,348,355,358,364,370,372,377,378,380],[312,310,310,310,310,312,315,318,323,327,334,341,347,353,357,359,364,366,368,370,370,370,369,369,367],[336,336,336,335,335,334,330,329,326,325,324,323,322,322],[272,271,271,269,268,266,265,265,265,266,270,272,280,284,296,300,305,315,324,333,342,349,352,355,357,362,363,365,366,366,366,365,364,361,360,354],[407,405,404,404,404,404,404,404,404,403,403,402,401,399,399,398,398,397,397,396,396,396,396],[422,421,423,429,437,448,459,464,473,477,481,485,487,491,494,495,496,496,494,493,487,485,476,474,472,470,466,465,465,469,472,476,478,480,487,491,494,497,500,501,502,502,502,502,498,493,487,483]],"t":[[0,20,28,31,36,48,53,65,70,82,98,103,115,120,132,137,148,154,165,171,182,187,195,204,215,220,220,228,236,253,254,265,270,271,281,287,298,304,315,320,331,337,348,354,365,370,378,387,399,403,415],[892,903,906,907,912,920,931,937,948,953,965,970,975],[1351,1370,1374,1379,1387,1398,1404,1415,1420,1432,1437,1437,1448,1451],[2097,2105,2115,2121,2132,2137,2149,2154,2165,2171,2182,2187,2198,2204,2204,2215],[2857,2863,2866,2871,2882,2887,2899,2904,2916,2921,2932,2938,2949,2954,2955,2966,2968,2971,2982,2989,2999,3005,3015,3021,3032,3038,3049,3055,3066,3071,3083,3088,3099,3104,3105,3116,3121,3132,3138,3139,3149,3155,3166,3171,3182,3189,3199,3206,3216,3221,3229,3246,3254,3255,3266,3271,3282,3288,3299,3306,3313],[3637,3650,3657,3666,3671,3683,3688,3688,3700,3704,3705,3716,3721,3722,3733,3738,3750,3755,3770],[4040,4074,4088,4099,4105,4113,4121,4133,4138,4139,4150,4155,4166,4171,4172,4183,4188,4188,4200,4213,4222,4232,4238,4249,4255,4266,4271,4272,4283,4288,4300,4305,4305,4316,4322,4333,4339,4349,4355,4366,4372,4383,4388,4399,4405,4416,4421,4422,4433],[5024,5031,5039,5049,5055,5066,5072,5083,5089,5100,5105,5117,5123,5133,5138,5150,5155,5167,5172,5172,5184,5188,5189,5200,5205,5206,5217,5222,5233,5238,5240,5250],[5433,5450,5457,5466,5472,5483,5488,5500,5505,5506,5517,5522,5534,5539,5550,5555,5567,5572,5584,5589,5600,5605,5606,5617,5622],[5951,5971,5975,5980,5982,5989,6001,6006,6017,6023,6034,6039,6050,6056],[6376,6381,6385,6391,6400,6406,6417,6422,6434,6439,6451,6456,6467,6472,6484,6489,6489,6501,6506,6517,6522,6534,6539,6539,6551,6556,6556,6568,6572,6573,6584,6589,6591,6601,6606,6617],[1649600380770,1649600380784,1649600380799,1649600380807,1649600380816,1649600380824,1649600380835,1649600380841,1649600380849,1649600380858,1649600380869,1649600380874,1649600380886,1649600380891,1649600380891,1649600380903,1649600380908,1649600380908,1649600380916,1649600380925,1649600380936,1649600380941,1649600380949],[1649600381230,1649600381250,1649600381274,1649600381285,1649600381291,1649600381303,1649600381307,1649600381319,1649600381324,1649600381325,1649600381337,1649600381341,1649600381341,1649600381353,1649600381358,1649600381369,1649600381374,1649600381386,1649600381402,1649600381408,1649600381420,1649600381424,1649600381433,1649600381441,1649600381441,1649600381453,1649600381458,1649600381458,1649600381470,1649600381474,1649600381487,1649600381491,1649600381492,1649600381504,1649600381508,1649600381520,1649600381525,1649600381536,1649600381549,1649600381550,1649600381558,1649600381561,1649600381570,1649600381574,1649600381586,1649600381591,1649600381599,1649600381608]],"version":"2.0.0"}

    where we observe that -ab{"x":[[142,140,139,141,143,146,148,155,163,172,181,186,195,198,205,211,213,215,218,220],[366,368,369,370,373,373,374,373,371,368,362,360,350,347,340,330,324,319,317,313,311,311,313,316,320,324,330,334,344,348,360,363,372,375,380,383,385,387,388,390,392,395,400,404,408],[287,294,299,306,314,319,333,338,350,361,373,385,401,411,421,428,435,440,442],[365,366,366,366,366,366,365,363,360,359,357,354,352,348,347,345,345,345,345,347,351,353,357,361,367,374,377,386,388,395,397,399,401,402,402,403,402,398,396,394,391,385,382,378,372,363,358,355,353,351,351]],"y":[[308,309,309,309,309,308,307,306,304,301,299,298,296,295,294,293,293,293,294,295],[216,214,212,207,199,196,191,186,181,178,174,174,176,177,182,192,199,207,211,219,226,231,236,238,238,238,235,234,228,226,220,218,217,217,220,224,228,233,238,242,245,247,249,249,248],[340,338,336,334,332,331,328,327,325,323,322,321,321,321,322,323,325,327,328],[384,388,392,394,401,406,416,427,439,445,451,462,468,484,488,500,503,505,508,510,509,508,506,502,499,494,493,492,492,493,494,496,499,501,504,509,512,521,525,528,531,537,539,542,546,548,547,544,541,528,522]],"t":[[0,15,21,23,29,32,32,43,49,61,66,77,82,83,94,99,107,107,117,124],[379,384,394,399,414,416,428,432,444,449,461,466,477,482,494,499,511,516,527,532,543,549,560,567,577,583,594,599,610,616,627,632,644,649,660,666,677,682,694,699,710,716,727,733,744],[1053,1070,1078,1085,1094,1099,1110,1116,1127,1133,1144,1149,1160,1167,1177,1183,1194,1200,1210],[1434,1442,1450,1461,1466,1467,1477,1484,1494,1499,1501,1511,1516,1527,1533,1544,1549,1550,1561,1566,1583,1594,1599,1611,1616,1627,1634,1644,1650,1661,1666,1667,1677,1683,1683,1694,1699,1711,1716,1717,1727,1733,1733,1744,1749,1761,1766,1779,1783,1794,1799]],"version":"2.0.0"} is the slope of the line as we found out in the previous section. And let us name -cb=d{"x":[[188,214,220,262,265,269,274,276,278,279,280],[397,392,370,365,361,352,342,334,330,320,317,315,315,318,323,326,329,338,342,358,363,375,381,392],[470,483,490,499,502,510,513,517,524,530,538,540],[469,479,483,486,491,499,507,516,524,529,537,541,548],[704,706,707,707,708,707,705,702,698,688,680,671,663,659,645,642,632,629,624,624,624,627,631,637,640,647,651,656,665,679,688,696,700,710,713,720,722,726,727,728,728,728,728,727,726,725,723,722,720,719,719,718,717,716,715,715,715,715,715,715,717,719,722,724,725,729,731,734],[286.49999999999994,287.49999999999994,294.49999999999994,297.49999999999994,304.49999999999994,308.49999999999994,312.49999999999994,322.49999999999994,331.49999999999994,341.49999999999994,353.49999999999994,364.49999999999994,375.49999999999994,387.49999999999994,399.49999999999994,409.49999999999994,420.49999999999994,429.49999999999994,437.49999999999994,444.49999999999994,450.49999999999994,454.49999999999994,455.49999999999994],[373.49999999999994,376.49999999999994,376.49999999999994,377.49999999999994,378.49999999999994,377.49999999999994,377.49999999999994,376.49999999999994,374.49999999999994,372.49999999999994,370.49999999999994,367.49999999999994,364.49999999999994,361.49999999999994,360.49999999999994,359.49999999999994,358.49999999999994,358.49999999999994,358.49999999999994,358.49999999999994,359.49999999999994,361.49999999999994,363.49999999999994,364.49999999999994,368.49999999999994,373.49999999999994,378.49999999999994,383.49999999999994,388.49999999999994,393.49999999999994,397.49999999999994,400.49999999999994,402.49999999999994,404.49999999999994,404.49999999999994,403.49999999999994,401.49999999999994,396.49999999999994,391.49999999999994,384.49999999999994,378.49999999999994,371.49999999999994,364.49999999999994,358.49999999999994,353.49999999999994,349.49999999999994,346.49999999999994]],"y":[[409,407,406,402,402,402,403,403,404,405,407],[339,339,347,350,353,360,369,379,384,400,410,419,423,434,440,443,445,449,451,454,455,455,455,455],[378,375,373,372,371,370,369,368,367,367,367,367],[415,412,411,410,409,407,404,402,400,399,397,397,396],[394,389,386,384,380,376,373,369,367,365,365,368,371,374,383,386,398,402,414,418,426,430,433,435,435,435,434,433,429,420,412,402,396,377,371,353,347,329,323,305,295,291,282,276,270,267,266,267,271,274,277,287,299,305,319,326,336,346,368,390,409,423,436,449,454,468,475,479],[500,500,498,497,495,494,492,490,488,485,483,481,479,478,476,475,474,472,471,471,470,469,469],[525,517,516,516,517,523,527,531,540,550,562,573,584,595,602,608,613,616,619,620,620,618,617,616,613,611,609,608,607,607,608,609,609,610,612,613,615,618,621,623,625,627,627,626,623,618,611]],"t":[[0,18,24,59,65,68,73,85,90,91,99],[538,562,567,574,576,585,591,601,607,618,624,635,640,651,664,668,670,674,685,690,702,707,708,718],[1073,1091,1099,1107,1118,1124,1124,1135,1141,1149,1157,1168],[1359,1379,1383,1391,1391,1401,1407,1418,1425,1432,1441,1441,1452],[1838,1841,1849,1858,1869,1874,1885,1891,1902,1908,1919,1925,1935,1942,1952,1958,1969,1974,1986,1991,2002,2008,2016,2024,2036,2041,2041,2052,2058,2069,2075,2085,2091,2102,2108,2119,2126,2135,2141,2152,2158,2169,2174,2186,2191,2199,2208,2219,2224,2233,2236,2241,2253,2258,2266,2270,2274,2275,2286,2291,2302,2308,2319,2324,2336,2341,2349,2359],[1649600391405,1649600391424,1649600391429,1649600391430,1649600391435,1649600391437,1649600391445,1649600391457,1649600391461,1649600391473,1649600391478,1649600391490,1649600391495,1649600391503,1649600391512,1649600391520,1649600391528,1649600391540,1649600391545,1649600391556,1649600391562,1649600391570,1649600391575],[1649600392030,1649600392049,1649600392062,1649600392072,1649600392079,1649600392095,1649600392099,1649600392105,1649600392112,1649600392120,1649600392129,1649600392137,1649600392145,1649600392154,1649600392162,1649600392173,1649600392179,1649600392187,1649600392195,1649600392204,1649600392212,1649600392220,1649600392224,1649600392228,1649600392237,1649600392245,1649600392254,1649600392262,1649600392270,1649600392278,1649600392290,1649600392295,1649600392306,1649600392312,1649600392323,1649600392328,1649600392340,1649600392345,1649600392357,1649600392362,1649600392373,1649600392379,1649600392390,1649600392395,1649600392406,1649600392412,1649600392424]],"version":"2.0.0"} where d is just another constant renamed in terms of c and b. Recall that -cb{"x":[[190,194,203,211,220,224,234,239,243,253,258,267,275,279,286,293,295],[441,438,437,435,431,428,421,414,405,401,392,388,385,379,375,373,373,378,380,383,389,393,397,402,417,422,432,437,442,452],[379,378,378,381,386,394,403,414,425,431,441,446,451,456,464,471,477,480,481,482],[438,438,438,438,438,437,436,435,433,432,430,429,428,426,425,422,422,421,421,421,421,421,422,423,426,430,435,441,447,455,461,466,468,473,474,475,475,473,471,465,462,459,453,450,448,443,441,438,436,436,436,436,437]],"y":[[308,308,308,308,307,307,306,306,305,304,304,303,303,302,302,302,302],[198,195,193,192,192,192,194,198,204,208,218,224,230,241,252,263,267,274,277,278,282,282,283,283,283,282,280,279,277,275],[352,352,353,353,353,352,351,349,347,346,344,343,342,341,340,340,340,340,341,344],[383,384,385,391,395,406,413,418,430,436,448,453,459,470,475,488,492,500,502,503,505,506,506,506,504,501,497,494,490,488,488,488,489,494,496,501,504,512,515,523,526,529,533,535,536,537,537,535,530,526,522,517,511]],"t":[[0,8,13,22,27,40,44,45,56,63,63,76,78,90,94,106,111],[770,775,779,786,795,795,806,811,822,828,836,839,845,856,861,873,878,889,895,895,906,911,912,923,935,939,945,945,956,961],[1150,1155,1164,1186,1195,1203,1211,1223,1228,1236,1245,1245,1253,1253,1262,1270,1278,1290,1296,1303],[1501,1520,1523,1529,1540,1545,1545,1556,1561,1562,1573,1578,1580,1590,1595,1606,1612,1623,1628,1628,1640,1645,1645,1653,1662,1670,1678,1690,1695,1706,1713,1720,1728,1740,1745,1757,1762,1773,1778,1787,1795,1796,1806,1811,1812,1820,1828,1837,1845,1846,1853,1857,1862]],"version":"2.0.0"} is the y-intercept itself, which here will be d. Hence, our equation is reduced to

    y=mslope x +dy-intercept

    Find the slope and the y-intercept of the line 3x-2y+1=0{"x":[[107,106,106,106,107,115,124,127,137,143,148,150,153,156,157,157,156,150,144,137,134,127,119,117,114,114,116,119,123,129,135,141,145,153,156,160,161,161,160,157,152,149,138,135,122,114],[225,225,225,226,227,228,233,236,247,254,263,270,273,282,286,288,289,290,288,285,283,274,271,268,265,257,256,250,249],[341,342,342,343,343,343,340,334,331,320,316,305,298,292,290,286,281,280,280,281,287,293,301,309,314,325,330,336,341],[417,419,422,426,429,439,443,454,457,466,468,471,474,476,477,478,479],[521,520,519,519,518,518,520,520,525,529,534,540,546,551,554,556,560,561,565,566,565,564,563,559,557,554,551,544,541,535,534,533,533,536,540,543,554,558,573,578,594,604],[658,660,661,662,663,663,663,663,663,663,661,661,660,660,658,658,658,661,663,670,676,681,688,692,703,706,716,719,728,731,733,737,740,741,741,743,743,743,742,740,738,736,734,733,730,727,726,724,721,720,718,718,717,716,716,717,718,720,723,724,726],[818,818,819,820,823,826,835,837,843,852,855,863,866,869,873,877,881,884,887],[858,858,858,857,856,856,855,855,855,855,855,855,855,855,856],[968,971,971,971,971,970,966,965,963,961,960,958,955,953,951,949,948,947,947,947,947,948],[1030,1033,1034,1036,1041,1047,1051,1062,1066,1079,1083,1087,1095,1099,1102,1107,1112,1114,1117],[1041,1044,1048,1050,1052,1058,1066,1070,1081,1086,1090,1097,1103,1110,1114,1118,1120],[1184,1183,1181,1179,1177,1175,1173,1173,1173,1176,1177,1184,1186,1191,1200,1206,1213,1216,1226,1228,1232,1238,1241,1244,1246,1252,1254,1255,1255,1254,1251,1248,1246,1242,1234,1230,1218,1214,1200,1191,1184,1181,1180]],"y":[[279,276,275,270,266,258,253,252,249,249,250,251,253,258,260,268,271,283,292,301,305,315,326,329,337,339,344,347,350,354,357,361,362,368,369,375,378,382,387,390,394,396,399,401,402,401],[315,312,310,307,305,302,298,295,289,286,285,285,286,291,297,302,306,318,327,338,342,357,361,367,370,380,383,389,391],[302,298,295,294,293,292,292,294,296,305,309,321,330,339,342,351,364,368,379,383,389,392,394,395,395,395,395,394,392],[332,332,333,334,334,335,335,335,335,334,334,333,332,332,332,332,332],[299,298,297,296,295,294,291,290,287,285,283,283,284,285,287,288,291,293,299,302,310,314,317,322,326,329,333,342,346,354,356,361,363,364,364,364,362,361,357,356,352,349],[288,286,285,285,285,287,289,297,301,305,314,319,325,329,339,343,347,354,355,358,358,358,356,355,348,345,335,331,318,314,310,303,299,297,295,289,287,284,284,284,287,291,297,300,312,321,331,337,355,361,372,378,389,394,408,412,422,427,432,433,434],[325,327,327,327,327,326,324,323,322,321,320,319,319,319,318,318,318,318,318],[286,287,288,294,302,311,321,325,337,339,348,352,357,360,362],[265,263,264,265,272,277,291,298,305,318,325,332,345,355,360,374,378,387,392,395,397,398],[291,293,293,293,293,293,293,292,292,292,292,292,292,292,292,292,292,292,293],[330,331,331,331,331,331,330,329,328,328,327,327,326,326,326,325,325],[278,285,294,305,312,331,343,349,365,382,385,394,397,399,401,402,399,398,389,386,382,373,368,363,358,341,334,317,311,295,284,273,268,261,254,252,253,255,263,270,278,282,285]],"t":[[0,19,22,23,37,63,67,80,84,92,101,109,119,126,137,142,154,169,169,176,184,193,203,209,220,226,237,242,251,259,270,276,287,292,304,309,320,326,337,342,354,359,370,376,387,393],[714,718,726,727,734,737,744,751,759,767,776,786,793,803,809,819,826,836,843,851,859,868,876,878,886,893,903,909,920],[1115,1122,1131,1134,1143,1151,1168,1176,1187,1193,1203,1209,1220,1226,1237,1243,1253,1259,1270,1276,1287,1293,1303,1309,1320,1326,1327,1337,1342],[1627,1668,1676,1687,1693,1703,1710,1720,1726,1737,1743,1743,1754,1759,1760,1771,1775],[2050,2050,2054,2060,2064,2068,2076,2087,2093,2103,2110,2120,2126,2137,2143,2144,2154,2160,2170,2176,2187,2193,2193,2204,2210,2210,2220,2226,2237,2243,2251,2260,2270,2276,2287,2293,2303,2310,2320,2326,2337,2344],[2593,2606,2612,2613,2618,2622,2626,2638,2643,2643,2654,2660,2660,2671,2677,2677,2687,2693,2704,2710,2720,2727,2737,2743,2754,2760,2771,2777,2788,2793,2793,2804,2810,2810,2821,2827,2837,2843,2854,2860,2868,2877,2887,2893,2904,2910,2921,2927,2938,2943,2952,2955,2960,2971,2977,2988,2993,3004,3010,3022,3034],[4983,4987,4994,5002,5013,5019,5037,5045,5053,5061,5069,5077,5086,5088,5094,5102,5111,5121,5136],[5348,5378,5386,5394,5404,5411,5422,5428,5438,5444,5455,5461,5472,5478,5488],[5865,5881,5887,5894,5903,5911,5922,5928,5928,5939,5944,5945,5956,5961,5972,5978,5989,5994,6005,6011,6022,6030],[6293,6296,6304,6304,6311,6321,6328,6339,6344,6356,6361,6361,6373,6378,6378,6389,6395,6405,6411],[6586,6590,6597,6603,6607,6611,6623,6628,6639,6645,6648,6656,6661,6672,6678,6689,6694],[6958,6970,6979,6987,6995,7005,7012,7020,7031,7048,7053,7063,7070,7078,7087,7095,7106,7112,7123,7128,7128,7140,7145,7145,7156,7162,7173,7178,7189,7195,7206,7212,7222,7228,7239,7245,7256,7262,7273,7278,7289,7295,7295]],"version":"2.0.0"}.

    Solution

    To compare the given equation of the line with the slope- intercept form, we need to solve it explicitly for y, we have

    2y=3x+1{"x":[[95,95,96,97,98,102,103,110,112,119,123,127,129,131,134,136,138,139,139,139,138,136,134,132,128,127,123,121,117,116,114,113,113,114,117,120,123,128,130,137,144,152,155,167,171,181,184,189],[224,225,227,228,229,229,228,227,227,226,227,229,231,234,238,240,247,249,257,259,262,267,270,276,280,282,286,287,289,290,290,290,290,289,287,287,285,284,282,281,281,280,279,278,278,278,278,278,278,278,277,276,276,274,273,270,269,265,262,261],[391,392,393,394,396,399,400,403,407,412,414,419,424,429,432,436,438,442,443,444],[397,396,396,397,398,401,408,410,416,425,431,433,438,445,448,451,452],[526,525,525,524,525,525,526,529,530,538,543,546,555,557,562,565,566,567,567,565,564,559,558,553,552,550,548,548,547,547,547,548,549,552,553,555,560,564,567,569,570,571,571,567,566,559,554,550,547,545,542,538,536,535,533],[631,629,629,629,629,630,634,635,642,647,652,655,663,664,666,668,672,674,676,676,676,673,672,666,664,658,656,654,651,649,647,646,645],[710,712,712,711,709,708,703,699,695,692,685,684,681,680,680,681,682,685,692,695,702,705,709,722,730,738,742],[789,790,791,792,795,799,804,807,816,819,828,831,834,839,843,847,849,850,852],[825,825,825,826,826,826,826,825,824,823,821,820,818,817,817,817,817,817,819,819],[914,918,919,920,921,923,924,924,924,924,922,921,919,918,916,914,914,914,915,915,917,917,918,919,920,921]],"y":[[310,309,308,307,306,303,301,298,296,294,294,294,295,297,299,302,306,308,313,318,324,330,336,338,347,349,356,359,364,366,370,371,373,374,374,374,373,372,371,368,366,363,362,358,357,354,353,351],[292,289,287,287,289,293,298,312,316,328,334,338,343,346,349,350,351,351,349,348,346,342,340,333,328,325,318,315,311,306,303,301,300,299,299,300,304,306,314,317,320,324,337,342,358,363,379,383,396,404,410,415,418,421,421,422,421,417,412,410],[305,305,305,305,305,305,304,304,304,303,303,303,304,304,305,305,305,306,306,307],[333,334,335,335,335,335,335,334,333,332,331,331,329,328,327,326,325],[290,289,288,287,285,284,283,280,279,276,274,273,273,273,274,276,278,280,282,287,289,295,297,302,304,306,308,309,311,313,314,315,316,318,318,319,322,324,325,328,329,332,333,336,337,339,341,342,342,342,342,341,340,339,337],[289,286,285,284,281,280,275,273,270,268,268,268,271,272,273,275,281,283,289,295,297,305,307,314,316,321,323,324,326,327,328,328,328],[267,265,264,264,264,264,266,268,272,274,283,287,298,301,304,310,313,317,322,323,325,325,325,325,323,321,320],[291,292,292,292,292,292,292,292,291,290,290,290,290,289,289,289,289,289,289],[266,265,264,264,265,266,270,275,278,282,290,295,307,311,314,320,324,326,331,332],[254,252,252,251,251,251,251,253,254,256,265,269,278,283,292,301,309,312,322,326,335,338,344,348,350,350]],"t":[[0,25,32,43,49,59,65,76,83,98,100,111,115,126,132,143,149,149,159,165,176,184,197,199,212,215,226,232,243,249,259,265,276,282,293,300,309,316,316,326,333,343,349,360,365,376,382,390],[650,659,674,691,699,709,716,737,740,749,759,766,776,782,793,799,809,816,826,832,832,843,849,859,866,874,882,892,899,909,916,926,932,943,958,966,976,982,993,999,999,1009,1016,1028,1032,1041,1049,1060,1066,1076,1083,1093,1099,1113,1116,1125,1132,1142,1149,1159],[1582,1587,1591,1599,1609,1616,1626,1633,1643,1649,1650,1661,1666,1676,1683,1693,1699,1713,1716,1724],[1868,1884,1888,1892,1899,1910,1916,1926,1933,1943,1949,1960,1966,1976,1983,1993,1999],[2293,2300,2310,2316,2327,2333,2333,2343,2349,2358,2366,2377,2383,2391,2400,2410,2416,2427,2433,2443,2449,2460,2466,2476,2483,2483,2494,2499,2500,2510,2516,2516,2527,2533,2533,2544,2550,2562,2566,2577,2583,2593,2600,2614,2616,2627,2633,2646,2650,2650,2661,2666,2667,2678,2681],[3026,3042,3045,3050,3060,3066,3077,3083,3094,3100,3110,3116,3127,3133,3134,3144,3150,3161,3167,3177,3183,3194,3200,3210,3216,3227,3233,3233,3244,3250,3250,3261,3264],[3420,3432,3442,3450,3460,3467,3477,3484,3494,3500,3510,3517,3527,3533,3534,3541,3550,3560,3567,3577,3583,3584,3594,3600,3611,3617,3625],[3884,3888,3895,3900,3909,3917,3927,3934,3943,3950,3960,3967,3970,3981,3984,3994,4000,4000,4011],[4174,4188,4194,4200,4225,4228,4235,4244,4250,4250,4261,4267,4278,4283,4284,4295,4300,4311,4317,4325],[4568,4578,4584,4585,4592,4600,4611,4617,4617,4628,4634,4645,4650,4651,4662,4667,4678,4684,4695,4700,4711,4717,4728,4734,4756,4758]],"version":"2.0.0"}

    Dividing by 2, we get,

    y=32x+12{"x":[[111,112,112,112,112,112,111,110,110,110,111,113,114,118,120,123,126,128,130,135,139,143,147,150,154,155,159,160,163,163,164,164,165,165,166,166,167,168,168,169,170,171,171,170,169,166,164,158,153,148,145,140,134,127,120,114,111],[275,277,279,280,284,286,292,295,301,304,310,312,318,319,322,325],[271,275,277,278,286,289,297,302,308,311,313,316,318],[410,411,411,412,413,415,417,419,421,426,428,434,438,441,442,444,444,444,441,440,436,433,430,427,425,425,425,426,427,432,433,438,439,441,443,444,445,446,447,447,446,446,445,442,440,435,430,424,421,412,409,401,399],[398,396,397,400,404,410,414,426,430,435,444,448,452,460,467,470,479,481,486,487,489,489],[435,435,434,433,432,432,432,432,433,435,437,440,441,445,446,448,449,449,449,448,442,439,437,434,432,430,429,429,430,431,434,438,443,445,454,458,468,472,475,482,485,488,491],[531,531,531,531,531,531,531,532,533,536,537,540,542,547,551,556,557,562,563,566,566,566,564,561,560,558,555,551,549,546,544,543,540,539],[580,580,581,581,581,581,579,578,576,574,573,571,568,567,565,564,563,564,565,566,571,573,579,582,592,599,606,610],[657,667,676,679,686,689,696,704,707,710,715,721,724,726],[697,699,700,701,702,702,701,700,697,696,694,693,693,693,693,694,695,697],[826,826,826,826,826,826,825,825,823,821,820,819,818,818,818,818,818,818],[778,779,780,783,787,789,793,801,809,814,823,828,835,842,849,855,859,860,863,864],[796,797,798,800,801,804,805,808,812,815,817,822,823,827,828,828,828,827,825,822,818,816,808,806,799,795,792,791,791,791,800,803,809,812,815,827,830,839,843,850]],"y":[[301,301,302,304,308,314,317,329,334,344,349,354,355,359,360,361,361,361,361,358,356,352,347,342,337,334,326,323,317,315,312,314,319,322,328,331,335,344,349,359,369,378,383,397,402,413,417,426,430,433,434,434,434,432,427,422,420],[311,311,311,311,311,311,310,310,309,309,309,309,309,309,309,309],[342,342,342,342,341,340,338,336,335,334,334,333,333],[290,287,285,284,281,278,276,274,272,269,268,267,266,266,266,268,269,272,278,280,286,290,294,299,302,305,307,311,312,316,317,321,323,325,328,329,330,332,335,336,338,339,340,342,343,345,346,347,347,347,347,345,344],[372,372,371,370,369,367,367,366,366,366,365,365,365,364,364,364,363,364,365,366,367,368],[417,418,418,417,416,415,414,413,413,412,412,412,412,415,416,420,422,427,429,431,440,443,445,448,451,453,453,454,454,454,454,454,454,454,452,451,449,448,447,445,444,443,442],[341,337,336,333,331,328,325,323,322,318,317,315,315,315,315,316,317,321,323,329,332,339,344,349,352,354,359,363,365,368,369,370,371,370],[325,324,324,323,322,321,321,321,322,325,327,329,334,337,343,349,354,359,363,364,369,370,372,372,371,369,366,364],[342,343,342,342,341,340,339,338,337,337,336,336,336,335],[323,323,323,323,324,326,333,336,348,352,363,369,373,377,380,382,383,384],[271,269,268,267,268,269,274,276,282,290,297,303,308,312,314,319,320,324],[368,368,368,368,368,368,367,366,364,364,363,362,362,361,361,361,361,361,361,361],[410,404,403,400,398,396,394,392,390,389,389,388,389,391,392,396,399,403,407,412,416,419,425,428,433,437,440,441,442,443,446,446,446,446,445,444,444,443,443,441]],"t":[[0,8,57,68,73,84,90,101,107,118,124,134,140,151,157,168,174,174,184,190,201,207,218,224,234,240,251,257,267,273,284,317,324,334,340,341,351,357,357,367,374,384,390,401,407,418,424,434,440,451,457,468,474,489,490,501,508],[975,983,990,991,1001,1007,1017,1024,1034,1041,1051,1057,1067,1074,1084,1091],[1296,1301,1304,1309,1316,1324,1335,1341,1351,1357,1357,1368,1370],[1677,1687,1692,1699,1708,1718,1724,1734,1741,1751,1757,1768,1778,1783,1791,1801,1807,1816,1824,1834,1841,1851,1857,1868,1874,1885,1891,1901,1907,1918,1924,1935,1941,1941,1951,1958,1958,1968,1974,1984,1991,1991,2001,2008,2008,2018,2025,2035,2042,2051,2058,2068,2074],[2425,2440,2458,2468,2474,2485,2491,2502,2508,2508,2518,2524,2526,2535,2541,2552,2558,2570,2574,2588,2591,2592],[2797,2816,2841,2851,2867,2869,2875,2891,2892,2902,2908,2918,2925,2935,2941,2952,2958,2968,2975,2975,2996,2999,3008,3018,3025,3035,3041,3042,3052,3058,3068,3075,3085,3091,3100,3108,3118,3125,3125,3135,3141,3142,3149],[3409,3415,3420,3425,3435,3441,3451,3458,3469,3475,3485,3491,3492,3506,3508,3519,3525,3535,3542,3552,3558,3570,3575,3585,3591,3592,3602,3608,3620,3625,3625,3636,3641,3658],[3814,3820,3824,3833,3842,3852,3869,3875,3886,3892,3892,3902,3908,3909,3919,3925,3936,3942,3953,3958,3969,3975,3986,3992,4003,4008,4019,4025],[4330,4346,4358,4368,4375,4376,4386,4392,4400,4403,4408,4417,4425,4435],[4602,4612,4618,4625,4642,4652,4658,4669,4676,4690,4692,4703,4709,4717,4725,4736,4742,4750],[5330,5340,5343,5351,5380,5384,5392,5403,5409,5420,5427,5436,5442,5453,5459,5470,5476,5486],[5805,5806,5817,5826,5834,5842,5843,5853,5859,5870,5876,5876,5887,5893,5904,5909,5920,5926,5937,5942],[6270,6281,6287,6292,6304,6309,6310,6321,6326,6336,6343,6353,6359,6370,6376,6387,6393,6403,6409,6420,6426,6437,6443,6455,6459,6470,6476,6485,6490,6493,6521,6526,6534,6535,6543,6551,6559,6570,6576,6587]],"version":"2.0.0"}

    Comparing this with the standard form y=mx+d{"x":[[93,92,92,91,91,91,92,92,93,95,97,98,99,102,104,110,113,120,123,130,133,139,143,145,148,149,151,153,154,154,154,154,154,154,154,153,153,151,150,148,143,141,138,132,129,125,120,118,111,106,104,102],[226,226,230,232,247,254,266,268,270,273,275],[233,232,231,233,237,239,244,254,258,270,277,283,288,293],[342,343,344,345,346,346,347,347,347,347,347,347,348,349,352,355,356,362,363,370,374,378,381,383,385,386,387,387,387,388,388,388,388,387,387,387,386,386,388,390,393,395,402,404,413,416,418,423,425,430,431,432,432,432,432,432,434,435,436,439,440,443],[490,489,489,489,489,491,495,500,504,507,515,524,527,533,535,537,540,541,542,543,543,541,539,535,531,529,524,517,515,508,507,506,505],[579,579,579,579,578,578,575,573,570,566,561,556,553,545,542,537,536,535,536,539,541,546,555,559,571,576],[642,645,647,649,657,661,667,671,674,684,691,697,702,705,711,713],[694,697,699,700,701,702,702,702,702,701,698,696,694,692,691,690,690,692,693,695],[822,819,815,813,809,807,802,800,798,793,791,789,786,784,780,779,779,779,781,788,793,798,801,810,814,817,820,831,834,839,842,846,850,853,854,856,857,857,857,855,854,851,849,848,846,839,837,833,831,829,825,824,823,822,822,822,826,827,829]],"y":[[256,257,258,259,262,266,279,286,300,313,323,327,330,334,336,336,336,331,329,321,318,308,300,295,287,282,274,267,265,268,270,284,290,312,321,347,354,379,388,402,424,430,440,451,453,454,451,448,437,425,419,412],[290,289,287,286,281,281,281,282,282,283,283],[325,326,327,327,327,327,327,326,326,323,321,318,316,315],[281,284,292,299,307,314,321,324,330,332,335,336,334,333,326,320,316,305,301,292,287,285,285,286,291,294,303,310,316,321,325,329,331,333,334,335,336,335,331,326,319,314,300,295,283,280,278,275,275,282,285,298,303,314,319,328,335,341,343,348,350,351],[282,279,278,277,276,275,274,273,273,272,271,271,271,274,276,279,285,288,291,298,302,309,315,322,328,331,336,343,344,347,347,347,344],[284,283,282,281,279,278,276,275,274,275,277,281,284,295,300,314,318,330,335,339,341,343,345,345,343,341],[304,302,301,300,299,299,298,298,298,298,298,298,298,298,298,298],[274,268,266,266,266,269,272,281,285,293,307,315,323,330,334,336,339,340,340,338],[281,278,276,275,273,273,273,273,274,276,278,280,285,287,298,305,308,314,317,320,319,316,314,307,303,300,295,281,276,265,259,250,241,232,228,217,214,207,205,203,203,204,206,208,212,227,234,248,256,265,284,292,307,319,328,332,341,343,345]],"t":[[0,8,15,23,24,34,40,40,53,57,68,73,74,84,90,100,107,117,123,133,140,153,157,168,173,173,184,190,200,217,223,233,240,253,257,268,273,284,290,300,307,317,323,334,340,349,357,365,373,383,390,395],[696,732,740,752,768,775,795,799,799,807,815],[963,967,974,1007,1015,1023,1034,1040,1051,1057,1070,1073,1085,1091],[1382,1382,1386,1392,1401,1407,1417,1424,1434,1440,1451,1467,1484,1490,1501,1507,1517,1524,1534,1541,1551,1557,1567,1574,1584,1591,1601,1607,1619,1624,1633,1640,1651,1657,1658,1666,1675,1691,1703,1707,1718,1724,1734,1741,1751,1757,1758,1768,1774,1795,1799,1807,1818,1824,1825,1834,1842,1851,1858,1868,1874,1884],[2165,2176,2182,2184,2193,2201,2207,2218,2224,2225,2236,2241,2253,2257,2258,2268,2274,2274,2287,2291,2295,2302,2307,2318,2324,2335,2341,2353,2358,2368,2374,2375,2383],[2535,2540,2543,2557,2558,2567,2574,2584,2592,2601,2608,2618,2624,2635,2641,2651,2658,2667,2674,2685,2691,2701,2707,2718,2724,2732],[2966,2970,2976,2985,2991,3005,3008,3008,3019,3024,3035,3041,3052,3058,3070,3074],[3251,3261,3267,3274,3285,3291,3302,3308,3318,3324,3335,3341,3352,3358,3368,3374,3385,3404,3410,3420],[3714,3720,3729,3736,3741,3752,3758,3758,3769,3775,3775,3786,3793,3794,3810,3816,3817,3826,3835,3852,3858,3869,3875,3885,3891,3892,3900,3908,3918,3925,3925,3936,3941,3952,3958,3969,3975,3985,3991,4002,4008,4019,4025,4025,4036,4041,4052,4058,4059,4066,4075,4076,4085,4091,4102,4108,4119,4125,4130]],"version":"2.0.0"}, where m is the slope and d is the y -intercept, we get m=32,d=12{"x":[[111,111,112,112,113,114,114,115,116,117,118,118,118,118,118,119,120,122,122,123,124,126,127,129,130,131,135,137,139,143,146,149,152,154,160,163,165,165,167,168,169,169,170,170,170,170,170,170,171,173,174,177,179,185,187,190,195,197,203,205,208,210,211,212,212,213,213,213,213,213,215],[285,287,288,290,298,301,308,311,315,324,330],[296,301,307,309,318,323,329,335,340,342],[409,410,411,411,412,414,415,417,418,422,423,426,427,430,432,433,434,434,434,434,432,430,429,428,427,427,427,428,429,433,438,446,451,456,460,463,464,465,465,461,459,453,448,442,439,430,424,421,416,412],[412,414,417,419,428,432,445,450,459,473,481,489],[436,435,435,435,437,439,444,446,452,455,461,465,469,471,472,473,473,471,470,467,465,461,456,445,444,443,443,446,450,455,457,467,471,478,482],[557,557,557,557,557,557,557,556,555,554,554],[718,721,722,723,724,724,724,723,716,710,708,706,700,698,690,687,685,682,680,678,677,677,677,680,682,685,687,692,698,704,707,717,721,730,732,739,741,743,745,747,748,749,750,750,750,750,749,748,747,745,742,739,738,734,733,730,729,728,728,729,731,732,733,735,737,738,739,739],[796,798,799,800,801,805,807,809,813,816,821,826,830,834,838,839,842,843,843],[798,800,804,807,811,816,821,827,832,836,841,844,847,849,850,851,851],[950,953,953,954,954,955,955,955,954,953,952,951,950,950,949,949,948,948,948,948],[920,921,923,928,930,937,944,948,964,971,977,984,988,989,990],[949,947,946,945,944,944,943,943,944,946,949,951,955,960,964,971,975,978,979,980,980,980,976,974,967,962,957,952,950,943,940,941,943,946,950,955,961,964,972,975,983,987,990,992,993]],"y":[[218,216,214,213,213,214,215,222,229,236,245,254,263,268,277,279,281,279,277,276,271,264,260,253,248,244,233,229,225,218,215,211,211,212,222,232,235,238,243,246,253,256,261,266,268,270,269,267,265,258,255,246,242,233,230,227,222,220,218,218,221,225,228,239,247,257,265,269,279,282,287],[225,225,225,224,222,221,220,219,219,218,218],[256,257,256,255,252,250,248,246,244,244],[178,173,172,171,170,168,166,164,163,161,161,161,161,162,163,165,167,170,174,176,183,187,192,194,199,201,202,204,204,205,206,207,208,210,212,215,216,220,222,226,227,231,234,235,236,238,238,238,236,234],[272,273,273,272,271,270,268,267,266,266,265,265],[331,330,329,326,324,323,320,319,318,317,318,320,323,326,327,331,336,343,346,351,353,359,364,377,378,380,381,381,381,381,380,378,377,375,374],[271,272,273,275,280,283,291,294,302,306,307],[241,238,237,235,234,233,229,226,220,219,218,218,219,220,224,226,228,234,237,243,249,252,254,259,259,260,260,258,257,254,252,245,242,230,226,212,207,203,195,191,188,181,179,174,169,168,167,166,165,165,167,170,173,182,186,201,206,225,232,247,255,261,264,269,273,275,276,277],[224,223,223,222,222,221,220,220,218,218,217,216,216,215,215,215,215,215,216],[243,243,243,242,241,240,240,239,239,239,238,238,238,238,238,238,239],[149,144,143,143,142,142,144,151,154,161,164,171,179,185,190,195,199,202,203,206],[250,250,250,249,249,248,247,246,243,242,241,240,240,240,241],[307,307,306,305,303,302,298,294,291,288,286,284,282,281,280,281,283,286,290,295,300,303,312,315,325,332,338,343,346,352,358,362,363,365,366,367,368,368,369,369,371,372,373,375,377]],"t":[[0,14,18,24,32,38,48,54,65,71,82,88,98,104,115,121,132,149,154,155,165,171,182,188,188,198,204,205,215,221,221,232,238,249,273,281,288,288,298,304,315,321,332,338,349,354,371,382,388,399,404,415,421,432,438,438,449,454,465,471,482,488,496,504,515,528,531,538,548,554,565],[891,906,909,914,921,932,938,938,948,955,965],[1224,1244,1255,1264,1271,1282,1288,1296,1305,1313],[1607,1623,1626,1628,1631,1638,1649,1660,1663,1673,1682,1688,1697,1705,1715,1721,1732,1738,1749,1755,1765,1771,1782,1788,1799,1805,1805,1815,1822,1832,1838,1849,1855,1865,1873,1880,1888,1899,1905,1915,1922,1932,1938,1949,1955,1965,1972,1982,1988,1997],[2410,2417,2423,2430,2439,2449,2455,2466,2472,2482,2489,2501],[2888,2906,2914,2922,2933,2939,2949,2956,2966,2972,2983,2989,2998,3005,3016,3022,3030,3039,3049,3055,3056,3065,3073,3100,3105,3115,3122,3133,3139,3149,3155,3166,3172,3183,3185],[3537,3537,3541,3547,3564,3572,3583,3589,3599,3606,3614],[4069,4074,4082,4089,4090,4100,4106,4117,4143,4147,4156,4158,4166,4172,4181,4189,4190,4200,4206,4216,4223,4223,4231,4239,4249,4256,4256,4267,4272,4283,4289,4300,4306,4317,4323,4333,4339,4340,4350,4356,4356,4367,4372,4383,4389,4390,4400,4406,4406,4417,4423,4433,4439,4448,4456,4466,4473,4483,4489,4500,4506,4517,4523,4533,4540,4540,4550,4554],[5078,5082,5090,5098,5098,5106,5107,5117,5123,5123,5134,5139,5150,5156,5167,5173,5184,5190,5193],[5453,5469,5482,5490,5501,5506,5517,5523,5534,5540,5550,5556,5568,5573,5584,5590,5604],[6101,6106,6111,6115,6123,6123,6148,6157,6167,6173,6174,6182,6190,6200,6207,6217,6223,6235,6240,6251],[6559,6583,6590,6601,6607,6618,6623,6624,6640,6651,6659,6676,6682,6690,6701],[6959,6973,6982,6990,7001,7007,7018,7023,7034,7040,7051,7057,7068,7073,7085,7090,7101,7107,7118,7123,7134,7140,7154,7157,7168,7174,7185,7190,7200,7207,7235,7240,7249,7257,7268,7274,7285,7290,7301,7307,7318,7324,7335,7340,7357]],"version":"2.0.0"}.

    Hence the slope is 32{"x":[[339,342,345,350,357,365,370,376,380,383,384,385,383,380,378,367,356,351,350,349,350,353,357,361,367,372,375,382,383,386,386,386,384,381,376,370,367,356,353,346,342,336,328],[316,316,318,325,335,348,355,373,379,396,401,416,421,432,436,443,444,445],[381,380,380,380,379,379,380,382,383,387,391,396,404,408,410,411,411,410,407,404,400,397,393,392,391,393,395,400,406,410,415,425,440,452,464,470,475]],"y":[[152,149,145,143,140,139,138,139,140,143,147,152,157,163,166,180,192,199,201,204,207,209,211,213,216,219,220,225,227,232,233,235,239,243,247,251,253,258,259,261,262,263,264],[319,320,320,320,318,316,315,313,313,310,310,307,306,304,304,304,304,305],[384,384,383,382,381,379,378,376,375,373,371,370,370,372,375,380,386,393,400,409,417,424,431,434,443,449,451,454,456,457,457,458,459,459,459,459,458]],"t":[[0,7,17,24,31,42,48,59,65,75,81,92,98,106,114,141,148,158,164,175,181,192,198,209,214,225,231,242,248,259,264,265,275,281,292,298,309,314,325,331,332,342,376],[600,606,631,640,648,658,665,675,681,691,698,706,715,723,731,741,748,758],[967,981,991,998,1008,1015,1025,1032,1042,1048,1059,1065,1075,1082,1092,1098,1109,1115,1125,1131,1142,1149,1158,1165,1181,1191,1198,1210,1215,1215,1225,1231,1243,1248,1261,1265,1269]],"version":"2.0.0"} and the y-intercept is 0,12.

    To find the x-intercept, we set y=0, and we solve for x, and in this case we get,

    3x+1=0 3x=-1 x=-13

    and thus the x-intercept is -13,0.

    Find the slope and the y-intercept of the linex+2y=0{"x":[[140,140,141,143,144,151,154,161,165,168,175,178,184,187,194,196,200,201,201,200,193,190,187,181,174,170,163,161,157,156],[217,218,218,218,217,215,214,210,208,203,200,197,192,189,187,183,182,182,185,187,195,199,212,222,233],[278,276,275,274,275,278,283,289,296,303,311,314,324,327,330,335,336,338,340,341],[320,319,319,318,317,315,312,310,308,306,305,305],[390,388,388,387,387,387,387,387,388,389,394,397,406,409,419,423,425,426,427,426,425,422,421,419,416,414,410,407,405,405,405,407,412,414,424,427,430,438,445,448,454],[476,476,476,476,476,476,476,475,473,472,472,472,472,473,476,480,482,486,489,494,496,501,507,513,519,524,528,534,537,539,539,539,538,535,531,525,522,519,514,512,509,507,506,506,506,506,507,508,509],[617,622,626,628,635,642,649,657,664,668,678,681,688,691,695,696],[606,609,616,619,628,633,644,655,664,669,683,687,690],[770,773,775,776,777,777,775,773,765,763,756,753,753,753,754,756,760,763,765,774,778,785,789,797,805,813,817,828,831,841,846,850,852,852,851,850,845,842,839,807,802,797,793,781,779,777]],"y":[[285,277,276,273,272,267,265,262,261,261,261,261,262,264,269,272,280,283,293,297,309,314,317,325,332,336,343,345,347,347],[257,256,254,253,253,256,258,266,269,277,282,286,295,299,303,315,322,328,333,335,338,339,340,339,336],[302,301,301,300,300,300,300,300,300,299,298,298,298,298,298,298,298,298,298,299],[278,277,276,276,278,283,290,298,304,317,320,329],[281,277,276,275,274,273,272,271,269,268,266,265,264,264,265,268,272,277,282,288,290,295,298,300,304,306,309,312,315,316,317,317,319,319,319,319,319,319,319,319,319],[271,270,268,267,266,265,268,271,281,290,295,305,309,312,317,321,322,322,322,321,320,318,313,308,302,294,290,280,275,272,271,275,278,287,300,315,324,334,355,364,381,396,405,409,419,420,422,424,424],[273,272,271,270,270,270,270,270,269,269,269,269,269,269,269,269],[314,314,313,313,312,312,311,310,310,309,309,309,309],[254,251,251,251,255,258,271,277,299,307,331,345,355,360,364,368,372,373,374,375,375,373,371,367,361,353,349,334,329,312,300,287,275,264,253,248,240,237,234,227,228,230,233,243,247,256]],"t":[[0,17,23,30,35,43,55,59,60,72,76,77,84,93,101,110,121,126,138,143,155,159,160,172,176,185,193,202,210,226],[415,420,426,438,452,460,472,476,489,493,494,505,510,510,522,526,538,543,555,560,572,578,588,593,605],[1187,1196,1203,1210,1237,1244,1255,1260,1272,1277,1288,1293,1305,1310,1313,1322,1327,1327,1339,1343],[1518,1534,1541,1544,1560,1568,1578,1588,1594,1602,1611,1622],[1994,2001,2005,2011,2013,2022,2024,2027,2039,2044,2055,2060,2072,2078,2089,2094,2105,2110,2119,2128,2138,2144,2144,2156,2160,2161,2172,2177,2189,2194,2206,2210,2222,2227,2239,2244,2244,2252,2260,2269,2277],[2475,2481,2488,2496,2505,2511,2522,2527,2539,2544,2552,2561,2561,2569,2577,2586,2590,2594,2606,2610,2611,2623,2627,2639,2644,2656,2661,2673,2677,2689,2694,2712,2719,2728,2739,2744,2744,2756,2760,2761,2773,2777,2789,2794,2806,2811,2813,2819,2831],[3323,3336,3339,3345,3356,3361,3373,3378,3389,3394,3406,3411,3419,3428,3439,3444],[3710,3717,3722,3728,3736,3736,3744,3756,3761,3773,3778,3790,3792],[4106,4112,4120,4128,4140,4145,4156,4161,4173,4178,4190,4194,4207,4211,4211,4219,4228,4228,4240,4244,4257,4261,4261,4273,4278,4290,4295,4307,4311,4324,4328,4340,4344,4353,4361,4373,4378,4378,4390,4414,4419,4422,4428,4440,4445,4453]],"version":"2.0.0"}.

    Solution

    Comparing the given equation to the general form ax+by+c=0{"x":[[202,201,197,193,190,185,180,173,165,155,150,133,128,112,104,99,97,97,97,99,101,106,110,122,126,139,149,157,162,176,180,193,197,206,212,217,219,221,225,251],[293,293,292,293,294,295,301,308,313,322,327,332,340,344,350,353,355,355,352,350,344,341,333,328,324,322,322,321],[385,385,385,385,384,384,383,382,380,380,376,375,371,370,369,370,371,372,374,379,382,385,397,406,417,423],[470,472,475,482,486,498,503,511,517,524],[508,505,505,505,505,505,506,506,508,509,509,510],[578,581,581,581,580,579,578,576,575,574,573,572,571,570,570,570,570,569,569,569,570,572,574,578,580,590,594,597,604,607,615,619,621,622,622,620,616,608,605,596,591,586,582,581,579,578],[642,643,644,644,645,645,645,646,646,647,648,649,651,654,656,660,664],[680,683,685,691,694,697,705,707,714,715,716,717,718,718,718,717,715,713,712,711,709,708,708],[770,777,780,787,795,798,802,809,812,819,824,829,831,836,837,838],[806,806,808,809,809,809,809,809,808,807,807,807],[930,933,932,929,925,922,915,911,906,895,891,882,875,870,868,868,868,873,876,885,888,893,901,906,911,921,926],[971,972,973,979,982,991,995,998,1005,1008,1010,1016,1021,1022,1025],[967,968,970,977,984,992,996,1008,1011,1022],[1075,1076,1076,1076,1076,1077,1076,1074,1073,1068,1066,1061,1060,1059,1059,1062,1066,1069,1074,1079,1085,1088,1094,1097,1101,1109,1113,1115,1117,1117,1118,1117,1113,1111,1102,1098,1084,1079,1074,1070,1066]],"y":[[264,259,253,248,242,236,231,229,229,232,236,255,263,292,315,337,346,354,360,369,371,372,372,371,369,361,353,344,339,323,317,303,300,295,295,299,302,306,315,353],[267,266,262,260,258,256,251,247,245,243,242,242,244,247,253,262,272,278,294,300,317,323,338,347,353,355,357,358],[261,260,257,255,254,256,259,265,273,278,295,302,320,326,329,336,339,341,343,344,344,344,340,336,330,327],[291,288,286,284,283,280,279,278,276,275],[242,238,239,241,247,252,262,275,288,299,308,312],[206,193,194,195,203,207,212,223,228,235,249,257,263,274,279,283,287,292,294,296,299,299,299,296,294,287,284,282,279,278,279,282,285,287,292,297,303,310,313,319,322,324,325,325,322,321],[264,265,266,268,270,277,286,291,296,305,309,312,315,318,319,319,317],[306,303,300,294,290,287,278,275,269,269,269,270,274,278,283,295,310,327,336,346,366,390,396],[274,269,269,267,266,266,265,265,265,265,265,265,265,265,265,265],[250,246,250,256,266,272,286,293,300,312,316,320],[264,250,248,246,246,246,250,253,257,268,274,286,301,315,328,338,343,350,351,352,352,351,348,346,343,337,334],[266,261,260,257,256,253,253,253,252,252,252,252,252,252,252],[300,301,302,302,302,301,301,300,300,299],[241,239,238,237,239,245,253,264,272,297,306,331,337,354,359,368,371,372,371,366,359,354,342,336,328,308,293,280,269,258,250,246,238,237,236,237,244,248,251,255,259]],"t":[[0,8,14,23,30,42,46,58,63,75,80,92,96,108,113,125,129,130,142,146,146,154,163,175,179,192,196,208,213,225,230,241,246,258,263,275,279,280,288,344],[552,557,564,574,579,580,591,596,608,613,613,625,629,630,641,647,658,663,674,680,691,696,708,713,725,730,730,741],[894,899,906,914,930,946,958,964,974,980,988,998,1008,1013,1013,1025,1030,1030,1042,1046,1047,1058,1063,1075,1081,1092],[1299,1314,1322,1330,1342,1349,1355,1363,1372,1380],[1566,1585,1589,1597,1605,1605,1613,1625,1630,1641,1647,1658],[1840,1860,1872,1880,1891,1897,1899,1909,1913,1914,1926,1930,1930,1942,1947,1947,1959,1963,1964,1972,1980,1992,1998,2005,2014,2028,2033,2035,2042,2047,2055,2063,2072,2080,2092,2097,2109,2114,2126,2138,2139,2148,2158,2163,2175,2178],[2324,2331,2350,2356,2356,2366,2375,2380,2380,2392,2397,2397,2409,2414,2414,2426,2431],[2452,2457,2467,2472,2476,2480,2492,2497,2509,2513,2514,2526,2530,2531,2543,2547,2559,2564,2564,2572,2580,2592,2598],[2953,2973,2981,2992,2997,2997,3009,3014,3014,3022,3032,3042,3048,3059,3064,3072],[3213,3232,3247,3259,3264,3276,3281,3281,3293,3297,3298,3310],[3538,3557,3566,3576,3581,3593,3597,3598,3610,3614,3615,3627,3631,3643,3648,3659,3664,3676,3681,3693,3697,3698,3710,3714,3715,3727,3731],[3938,3956,3964,3976,3981,3989,3997,3998,4010,4014,4015,4027,4031,4043,4047],[4188,4207,4215,4226,4231,4239,4248,4259,4265,4277],[4466,4474,4481,4482,4498,4506,4515,4526,4531,4543,4548,4561,4565,4576,4581,4589,4598,4610,4614,4623,4631,4643,4648,4648,4656,4664,4676,4681,4694,4698,4710,4715,4727,4731,4743,4748,4760,4764,4765,4777,4780]],"version":"2.0.0"}, we get a=1,b=2,c=0{"x":[[87,87,88,88,88,90,90,91,91,91,90,88,86,81,79,71,68,58,51,43,40,33,29,27,27,26,26,26,26,26,28,29,32,36,41,44,53,56,66,95,97,100,104,105,107,107,107,107,107,107,108,110,112,116,120],[177,178,187,193,197,204,211,214,220,229,231,234],[168,169,170,171,174,177,184,193,201,205,209,213,220,223,226],[297,299,301,302,302,303,303,303,302,301,299,298,297,297,297,297,296,296,296,296],[341,343,344,344,344,344,344,344,343,342,341,340],[442,443,443,443,443,443,443,443,443,443,442,442,441,440,439,438,438,437,436,436,436,436,437,440,441,446,448,453,456,458,462,464,466,468,471,472,474,476,476,477,477,476,475,474,468,466,461,457,454,447,443,440,431,427,426,426],[528,530,531,533,539,545,551,558,560,568,571,577,578,581],[525,531,539,547,555,563,571,578,583,588,591,593],[628,634,635,650,655,660,662,665,669,670,671,671,670,669,668,665,662,661,656,655,652,651,650,649,649,650,653,657,659,668,670,680,683,686,691,696,700,701,702,703],[739,739,739,739,739,739,739,739,739,739],[864,857,853,850,847,837,834,824,818,813,811,810,810,810,811,817,823,830,838,842,856,860,873,877],[918,927,929,939,942,946,953,957,960,966,972,974],[925,925,923,924,928,930,933,936,943,951,956,970,974,978],[1040,1039,1037,1035,1033,1029,1028,1025,1024,1023,1023,1025,1026,1029,1034,1036,1044,1046,1054,1061,1066,1072,1075,1082,1084,1089,1090,1092,1092,1091,1088,1084,1081,1075,1066,1064,1058,1055,1050]],"y":[[271,270,269,266,265,258,255,250,247,242,238,233,231,225,224,222,222,228,236,247,253,273,288,299,305,310,315,323,327,329,334,335,336,336,333,331,322,318,305,275,276,278,287,291,303,307,323,335,341,352,361,370,372,375,375],[248,248,244,242,241,239,237,236,235,234,233,233],[293,293,293,293,292,291,288,286,283,282,280,279,277,276,275],[194,191,196,199,206,211,222,235,247,260,271,281,289,293,302,305,311,313,314,316],[313,316,317,318,324,327,331,335,339,343,345,349],[184,185,187,191,195,205,211,217,229,236,241,252,263,269,278,282,287,297,302,305,307,308,307,303,302,297,295,292,291,291,291,291,292,293,295,297,299,302,304,306,310,313,317,320,327,329,333,335,337,340,341,342,342,340,335,328],[267,266,266,265,263,261,259,257,256,255,254,254,254,255],[303,302,300,298,295,293,291,290,289,288,288,287],[233,222,220,210,209,209,209,210,214,216,223,226,235,238,242,250,256,259,269,271,276,278,279,283,284,285,286,286,286,284,284,281,281,280,279,278,278,278,278,279],[305,313,315,321,324,327,334,338,342,343],[196,197,201,204,207,218,222,236,246,257,262,267,271,283,286,295,299,303,306,307,308,308,306,305],[240,238,237,235,234,234,232,232,232,232,232,232],[276,277,279,279,280,280,279,279,278,276,275,272,271,270],[205,209,217,227,233,250,256,276,282,300,310,317,320,326,331,332,333,331,325,317,309,299,293,276,270,253,247,230,219,210,201,196,194,192,193,194,198,200,205]],"t":[[0,6,18,27,38,42,55,59,60,71,76,88,93,105,109,121,126,138,143,155,159,171,176,188,193,193,205,209,209,222,226,226,234,243,255,259,272,276,289,331,338,343,362,368,376,388,393,405,409,422,426,438,443,455,459],[781,801,805,807,812,821,827,838,843,851,860,868],[1046,1064,1068,1076,1080,1088,1093,1101,1110,1110,1118,1122,1126,1138,1140],[1379,1379,1393,1405,1410,1410,1422,1427,1438,1443,1455,1460,1472,1476,1488,1493,1505,1510,1510,1518],[1670,1690,1694,1702,1710,1721,1727,1738,1743,1755,1760,1772],[2042,2062,2069,2077,2080,2089,2093,2094,2105,2110,2110,2122,2127,2139,2143,2144,2152,2160,2168,2178,2185,2194,2210,2221,2227,2239,2243,2256,2260,2260,2272,2277,2277,2289,2293,2294,2306,2310,2311,2322,2327,2327,2339,2344,2355,2363,2369,2369,2377,2385,2388,2393,2405,2410,2422,2427],[2738,2757,2761,2769,2777,2788,2794,2805,2810,2822,2827,2839,2843,2856],[3013,3032,3044,3054,3060,3069,3078,3089,3094,3106,3110,3114],[3326,3345,3353,3381,3388,3394,3404,3412,3422,3427,3439,3444,3456,3460,3461,3473,3477,3489,3494,3502,3511,3511,3523,3527,3528,3539,3544,3552,3561,3573,3577,3589,3594,3594,3607,3610,3622,3627,3628,3640],[3860,3870,3878,3886,3887,3895,3902,3911,3919,3927],[4284,4295,4299,4305,4313,4327,4330,4339,4344,4356,4364,4364,4376,4378,4386,4394,4406,4411,4423,4428,4440,4444,4456,4461],[4735,4753,4761,4773,4778,4778,4786,4794,4795,4807,4811,4823],[4969,4987,4991,4996,5011,5012,5019,5024,5028,5040,5045,5057,5061,5066],[5284,5303,5313,5323,5328,5340,5345,5357,5365,5377,5378,5387,5395,5406,5411,5423,5428,5440,5445,5457,5461,5473,5478,5490,5494,5507,5511,5523,5528,5540,5545,5557,5561,5573,5578,5590,5595,5595,5607]],"version":"2.0.0"}.

    The slope-intercept form is given by y=-abx-cb{"x":[[97,94,94,94,94,94,94,95,97,100,102,106,108,110,115,118,123,129,134,137,147,150,160,162,171,175,179,181,182,182,182,179,178,177,176,175,174,173,172,170,169,168,167,166,166,166,166,166,166,167,168,170],[237,238,240,242,253,257,270,279,286,292,297,301,302,304,305,306,306,306],[241,240,241,242,250,253,258,268,273,278,287,296,300,307,311,314,320,321],[381,380,381,383,384,389,399,407,414,419,425,428,431,438,440,443,447,448,450],[529,529,530,532,533,535,537,539,541,541,542,542,542,540,537,534,532,522,519,515,510,505,503,498,497,497,498,500,504,508,511,514,517,524,528,532,539,543,545,550,552,554,554,555,555,555,556,559,561,564],[482,482,486,492,501,506,512,517,527,533,538,549,554,558,567,572,579,583,586,591,593,594,596,596,596],[529,529,529,526,525,521,520,517,515,514,513,513,514,514,516,518,521,525,530,535,537,545,547,553,555,560,561,562,564,564,564,564,564,564,563,560,558,554,548,543,536,530,527,521,519],[629,630,631,633,634,636,643,646,650,657,660,663,668,670,672,674,676,676,676,675,672,669,665,661,659,654,650,647,646],[707,707,706,706,704,703,702,699,698,696,694,690,688,686,682,681,680,679,681,682,689,693,704,708,724,734,745],[796,803,805,808,815,830,837,840,850,852,857,859],[982,982,983,984,984,985,985,984,982,980,979,975,973,965,963,956,953,951,949,946,945,945,945,946,948,949,951,955,958,963,969,976,980,990,993,1000],[927,928,937,940,952,956,970,975,988,992,1004,1009,1011,1013],[961,962,963,963,964,964,964,964,963,962,961,959,958,954,952,949,948,947,947,946,945,945,945,946,947,950,954,958,960,962,967,970,978,981,983,987,989,990,992,992,993,993,992,991,988,984,981,977,973,969,968,967]],"y":[[213,210,211,212,222,227,243,253,263,270,273,278,281,282,284,285,286,286,284,282,274,271,259,254,239,228,220,215,213,214,215,225,230,241,247,255,269,277,296,315,332,340,361,372,381,386,395,398,399,401,401,399],[266,266,265,265,264,263,261,260,259,258,258,258,258,259,259,260,261,262],[298,298,297,297,295,295,294,292,290,289,287,285,284,283,283,282,282,282],[255,253,253,253,252,252,252,252,251,251,251,250,250,249,249,249,248,248,248],[230,228,227,223,222,218,213,211,206,203,201,198,194,191,190,190,190,194,196,199,205,212,215,226,228,235,237,237,237,236,235,234,233,229,227,225,221,220,219,218,219,221,223,231,234,243,245,251,252,252],[306,303,303,302,301,301,300,300,299,299,298,297,297,297,296,295,295,295,295,295,295,295,297,298,299],[353,354,355,366,372,390,395,410,419,427,433,436,440,441,442,441,439,435,430,424,422,417,416,415,415,418,419,421,424,426,427,429,432,433,435,437,438,441,443,445,446,446,446,443,440],[263,261,257,255,254,253,251,250,250,250,250,251,254,257,259,264,269,271,277,279,284,289,294,298,300,303,306,308,308],[256,254,252,251,251,251,252,255,257,259,261,267,270,274,281,285,290,295,298,300,302,302,303,303,300,298,295],[275,274,274,274,273,271,271,271,271,271,271,271],[198,197,196,194,192,191,189,187,186,186,186,187,188,191,193,199,202,204,206,210,212,215,220,222,225,227,228,230,231,233,234,234,234,234,234,234],[292,292,292,292,292,292,291,291,291,291,291,291,291,291],[323,323,323,324,327,332,339,342,345,352,356,368,372,384,390,395,397,399,400,403,404,405,406,406,406,404,402,399,398,397,395,394,393,393,393,393,394,395,398,401,403,407,408,410,414,416,417,418,418,417,415,411]],"t":[[0,16,22,28,40,44,56,60,73,77,89,94,94,106,110,111,123,128,140,144,156,160,173,177,190,194,206,210,223,227,240,244,252,260,261,273,277,278,290,294,306,311,323,327,339,344,356,360,361,373,377,378],[843,877,889,895,906,911,923,927,939,944,952,961,973,977,978,989,994,994],[1194,1211,1221,1227,1240,1244,1244,1253,1261,1261,1272,1278,1289,1294,1294,1306,1311,1311],[1777,1797,1801,1806,1813,1823,1828,1839,1844,1856,1861,1861,1873,1878,1878,1890,1894,1895,1902],[2386,2405,2408,2420,2429,2440,2445,2445,2457,2461,2462,2473,2478,2486,2498,2498,2510,2512,2520,2520,2529,2540,2545,2557,2562,2573,2578,2590,2595,2603,2611,2612,2623,2628,2628,2640,2645,2645,2657,2661,2662,2673,2678,2690,2695,2707,2712,2723,2729,2736],[3014,3031,3039,3047,3057,3062,3062,3074,3078,3079,3090,3095,3095,3107,3112,3112,3124,3128,3130,3140,3145,3148,3157,3162,3166],[3406,3437,3445,3457,3462,3474,3479,3487,3499,3511,3512,3521,3528,3540,3545,3553,3563,3573,3579,3591,3595,3604,3613,3623,3629,3641,3645,3646,3657,3662,3662,3674,3679,3679,3691,3695,3696,3708,3712,3724,3729,3741,3745,3757,3762],[4320,4320,4324,4337,4338,4345,4357,4362,4362,4374,4379,4379,4391,4395,4396,4408,4412,4424,4429,4429,4441,4446,4458,4462,4474,4479,4491,4495,4503],[4656,4663,4671,4679,4691,4695,4696,4708,4712,4712,4725,4729,4729,4741,4746,4746,4758,4763,4774,4779,4791,4796,4808,4812,4825,4829,4837],[5466,5479,5487,5490,5496,5515,5521,5531,5541,5546,5558,5563],[6656,6662,6668,6675,6680,6692,6697,6713,6725,6730,6732,6743,6747,6759,6763,6775,6780,6780,6792,6796,6797,6809,6813,6826,6830,6830,6838,6847,6847,6859,6863,6875,6880,6892,6897,6904],[7296,7306,7310,7316,7325,7332,7338,7348,7358,7363,7376,7380,7393,7395],[7577,7589,7597,7597,7609,7614,7626,7630,7632,7643,7647,7659,7664,7676,7680,7692,7697,7697,7710,7713,7714,7726,7730,7742,7747,7759,7764,7776,7780,7783,7793,7802,7809,7813,7814,7827,7830,7831,7843,7847,7859,7864,7864,7876,7880,7893,7897,7909,7914,7926,7930,7938]],"version":"2.0.0"}, which gives us y=-x2{"x":[[152,152,153,153,153,153,152,151,150,150,152,153,159,161,169,175,181,188,191,197,201,204,211,214,217,243,244,245,245,246,246,246,245,244,242,239,237,235,234,233,231,228,226,224,223,221,220,217,216,210,207,197,190,183,179,175,167,163,160,156,149,147,143],[349,349,350,351,355,359,365,371,379,382,392,395,404,406,408,411,412],[337,342,350,358,366,375,383,391,395,401,404,407,414,416,421],[485,488,489,495,498,502,509,512,515,522,525,528,531,537,539,541,545,548,549],[621,620,619,618,618,618,619,620,623,625,627,632,635,639,642,648,652,655,661,664,666,669,670,671,671,670,667,663,661,653,650,643,642,641,639],[699,699,700,700,700,700,698,695,691,684,679,674,671,664,663,661,661,663,664,668,674,682,689,698,708,717,725,732],[609,607,606,605,606,610,619,631,637,642,655,661,679,685,694,702,709,714,718,721,721,721],[659,658,657,655,654,652,651,650,649,649,649,651,655,658,662,677,682,685,688,689,690,689,688,682,677,670,663,656,649,643,639,637,636,637,641,647,652,656,666,678,691,704,716,728,743,748,755]],"y":[[227,228,229,235,244,256,270,284,296,302,318,322,333,336,343,346,347,347,346,343,340,338,331,327,323,281,279,277,276,274,273,272,272,272,275,280,288,293,298,303,315,338,353,369,378,399,406,426,432,448,452,462,466,467,467,466,462,459,455,451,440,435,423],[276,277,277,277,276,275,274,274,273,272,272,272,271,271,271,271,271],[313,313,312,310,308,306,304,303,302,301,301,300,300,299,299],[262,262,262,262,262,261,261,261,261,261,261,261,261,261,261,261,261,261,261],[230,229,226,225,222,221,217,215,209,207,206,203,202,200,200,199,199,199,200,202,204,208,211,216,222,228,234,240,243,251,253,260,262,263,265],[216,215,213,212,211,210,210,211,213,219,223,229,232,241,244,253,255,262,264,267,268,269,269,269,267,265,263,261],[326,326,326,326,326,326,325,324,323,322,320,319,317,316,315,315,314,314,315,315,316,317],[398,398,398,398,398,396,394,392,389,386,382,379,375,373,371,372,375,379,384,390,395,401,404,413,418,424,430,435,440,443,447,450,451,451,451,451,450,450,448,447,445,443,442,442,441,440,439]],"t":[[0,7,39,44,56,61,73,78,90,94,107,111,123,128,140,144,157,161,173,178,178,190,194,195,203,252,254,261,264,273,278,278,290,294,303,311,324,328,335,340,344,357,362,374,378,390,394,407,411,424,428,437,445,453,461,462,473,478,478,490,494,495,507],[908,919,928,945,953,962,970,978,989,996,1006,1012,1023,1029,1031,1040,1045],[1218,1237,1245,1257,1262,1273,1278,1290,1295,1303,1307,1311,1323,1329,1340],[1851,1887,1896,1907,1912,1912,1924,1928,1929,1941,1945,1945,1958,1962,1962,1974,1978,1990,1995],[2594,2611,2615,2616,2622,2629,2640,2646,2657,2662,2662,2674,2679,2679,2691,2695,2696,2707,2712,2712,2724,2729,2729,2740,2746,2757,2762,2774,2779,2790,2796,2807,2812,2812,2831],[2986,2992,2996,2998,3004,3012,3029,3040,3045,3057,3062,3074,3079,3091,3096,3107,3113,3124,3129,3137,3146,3157,3162,3174,3179,3191,3195,3208],[3475,3485,3488,3494,3507,3512,3529,3541,3546,3549,3558,3562,3574,3581,3591,3596,3608,3612,3624,3629,3629,3641],[3893,3906,3912,3924,3929,3941,3946,3958,3962,3971,3979,3991,3996,4008,4012,4038,4046,4057,4063,4074,4079,4087,4096,4108,4113,4125,4129,4142,4146,4158,4163,4175,4179,4191,4196,4208,4213,4213,4225,4229,4241,4246,4258,4263,4275,4281,4292]],"version":"2.0.0"}.

    Thus the slope is -12 and the y-intercept is (0,0).

    In order to find the x-intercept, we set y=0 and solve for x. Thus, we get

    x=0

    and hence the x-intercept is (0,0).

    Gradient and Intercepts - Key takeaways

    • The non-zero coordinates of the point where a straight line intersects the two axes are known as the intercepts of that line.
    • For a line given by ax+by+c=0{"x":[[220,220,152,126,132,144,148,160,165,170,180,190,201,212,217,231,243,245,250,252,255,258,259,262,264,268,270],[286,289,291,293,297,300,306,309,315,318,324,327,332,334,337,338,338,337,333,328,322,316,313,306,304,302],[365,364,363,362,359,353,351,345,343,338,335,332,331,330,330,330,332,339,342,345,353,363,374,384,390],[435,437,438,443,446,452,456,460,463,467,474,478,481,492,499],[488,488,488,487,487,485,483,480,476,475,472,471,470],[547,549,553,556,557,559,560,561,561,561,559,558,555,552,550,550,549,549,550,550,554,557,560,565,572,575,581,584,588,596,599,601,602,601,600,597,594,592,586,583,575,573,566,561,558,554],[624,625,626,626,626,626,627,628,632,634,635,639,641,643,647,652,655,660,662,664,669,672,677,682,686,689,690,691,691,690,686,685,684,682,681,680,679,678,678,678,679],[743,744,744,747,752,755,758,762,769,776,784,789,791],[773,773,774,775,776,777,777,776,775,773,769,768,767,764],[877,876,876,875,874,871,869,867,860,856,851,841,827,823,815,814,812,813,819,822,825,831,835,838,847,855,859,869,878],[933,934,937,940,944,946,950,963,966,975,980],[937,937,941,943,955,959,964,973,982,986,992],[1038,1041,1040,1038,1037,1033,1027,1025,1019,1018,1017,1017,1019,1020,1022,1026,1029,1032,1037,1040,1044,1053,1059,1065,1071,1076,1081,1083,1084,1083,1081,1076,1072,1068,1060,1051,1047,1040,1036,1033,1030]],"y":[[258,255,254,320,328,326,324,315,312,308,299,290,280,271,267,260,265,268,276,279,289,298,304,311,313,316,317],[267,262,260,258,254,253,251,251,252,254,258,261,267,271,279,287,291,301,310,320,331,339,343,354,356,360],[250,249,247,247,248,255,257,269,274,285,295,310,319,324,333,336,338,343,344,344,343,340,335,331,328],[313,312,310,308,307,306,305,304,304,303,302,302,301,299,297],[276,274,269,269,272,280,290,302,317,324,341,346,353],[228,224,215,207,205,203,203,204,212,218,230,237,251,267,283,290,304,308,313,314,315,313,310,306,301,298,294,292,292,294,298,305,308,315,319,327,331,334,340,343,350,352,354,355,355,353],[275,277,280,283,298,304,319,324,332,333,334,335,335,335,333,329,326,321,318,315,309,306,300,294,289,287,287,288,300,305,327,335,344,364,373,382,390,405,412,417,425],[302,302,301,301,300,299,299,299,298,298,298,298,298],[288,286,282,281,278,277,278,283,286,291,307,313,319,332],[270,269,267,264,263,261,260,260,262,264,268,277,297,305,327,334,352,356,364,365,366,366,366,366,364,360,358,353,347],[291,291,290,289,288,288,288,285,285,284,284],[323,324,325,325,323,322,321,319,317,316,314],[266,263,267,274,279,291,316,326,352,359,366,372,382,385,387,389,390,390,388,385,383,371,361,349,335,320,303,293,271,265,259,250,247,245,244,246,248,254,258,262,268]],"t":[[0,18,44,78,93,107,114,123,131,133,143,148,159,164,176,187,198,211,214,215,226,231,243,248,249,259,265],[549,561,565,565,576,581,590,597,599,610,615,615,626,631,643,648,649,659,666,676,683,690,698,710,715,726],[883,895,903,907,915,932,940,948,960,965,976,981,995,1002,1006,1007,1016,1026,1032,1033,1043,1048,1059,1065,1076],[1249,1257,1258,1265,1273,1282,1282,1290,1290,1304,1307,1307,1315,1326,1331],[1499,1506,1520,1525,1540,1549,1560,1565,1576,1582,1593,1602,1614],[1779,1784,1794,1804,1812,1815,1815,1823,1832,1840,1848,1849,1860,1865,1877,1882,1893,1904,1907,1916,1933,1940,1949,1960,1965,1977,1982,1982,1994,2004,2014,2017,2027,2032,2032,2044,2048,2051,2060,2065,2077,2082,2093,2104,2113,2116],[2313,2316,2324,2332,2343,2349,2360,2365,2377,2382,2382,2394,2399,2399,2407,2415,2428,2432,2432,2444,2449,2449,2461,2465,2477,2482,2494,2499,2513,2516,2528,2532,2532,2544,2549,2549,2561,2565,2566,2577,2582],[2817,2832,2843,2849,2860,2865,2866,2878,2882,2894,2899,2911,2918],[3063,3068,3078,3082,3091,3091,3120,3124,3124,3132,3144,3149,3153,3161],[3455,3460,3464,3469,3478,3482,3483,3495,3499,3499,3511,3516,3527,3533,3541,3549,3560,3566,3577,3582,3583,3595,3599,3599,3607,3616,3624,3632,3644],[3880,3888,3894,3899,3911,3916,3916,3933,3941,3949,3961],[4127,4145,4158,4166,4177,4183,4183,4195,4199,4211,4216],[4401,4421,4424,4433,4441,4449,4461,4467,4478,4483,4483,4495,4499,4500,4512,4516,4516,4528,4533,4533,4545,4550,4561,4566,4578,4583,4595,4600,4611,4616,4616,4629,4633,4633,4645,4649,4662,4666,4667,4678,4683]],"version":"2.0.0"}, the x-intercept is given by -ca{"x":[[211,214,216,222,227,240,244,249,254,264,269,275,287,292,298,309,315,326,337,346,350,359,361,365,366],[489,490,491,492,493,494,496,496,494,492,490,488,478,474,461,456,447,441,437,429,419,416,415,415,416,419,423,430,433,438,447,452,458,468,473,479,484,494,498],[410,412,413,415,421,432,441,446,455,460,465,474,487,495,501,506,510,511,512,512],[479,483,485,487,489,490,490,491,491,490,487,484,482,480,474,472,464,461,458,456,453,451,450,448,448,448,448,451,453,459,461,469,472,482,489,497,500,503,508,513,514,517,520,522,525,529,534,536,546,549,553,560]],"y":[[279,279,279,278,278,276,275,274,273,272,271,271,270,270,269,269,269,269,268,268,268,269,269,271,271],[180,179,177,176,175,174,170,169,166,165,164,164,165,166,173,176,184,188,193,203,220,228,236,239,243,248,252,256,257,259,260,261,261,261,260,260,259,258,257],[331,331,331,331,330,328,326,325,323,321,320,318,316,315,315,315,315,316,318,320],[396,397,395,393,390,388,386,381,379,372,369,366,366,366,366,366,372,374,377,381,388,393,397,407,411,415,421,426,426,427,427,425,423,416,410,404,402,400,399,401,403,410,416,421,426,429,429,429,426,425,422,417]],"t":[[0,6,11,23,27,39,43,43,55,59,60,72,76,76,89,93,93,105,110,122,126,139,143,155,159],[436,445,447,447,455,460,472,477,489,493,493,505,510,522,526,538,543,544,555,560,571,577,588,593,596,605,614,621,626,627,638,643,643,651,660,660,671,676,680],[920,927,929,938,943,955,960,971,977,977,988,993,1005,1010,1021,1027,1038,1044,1055,1060],[1282,1297,1305,1313,1321,1327,1329,1338,1343,1355,1360,1371,1377,1377,1388,1393,1405,1410,1410,1422,1427,1427,1439,1443,1444,1455,1460,1472,1477,1489,1494,1505,1510,1522,1527,1539,1543,1544,1555,1560,1572,1577,1588,1594,1605,1610,1622,1627,1638,1643,1646,1655]],"version":"2.0.0"} and the y-intercept as -cb{"x":[[342,347,349,352,360,364,369,383,392,400,408,412,422,425,431,433,436,437,438],[591,590,589,585,579,573,566,559,552,544,538,535,528,527,526,527,528,531,535,538,545,553,567,572,586],[512,530,535,540,546,559,571,582,594,600,615,620,631,633,639,640],[557,556,555,555,554,553,553,552,550,549,548,547,544,542,541,540,540,541,546,548,555,557,567,570,580,586,592,594,599,601,603,603,602,600,594,588,581,577,567,563,560,554,552,550,546,545]],"y":[[301,300,299,299,297,296,295,293,291,290,289,289,289,289,290,291,293,294,295],[195,193,192,191,192,195,200,205,212,221,230,235,250,254,265,268,271,276,279,281,283,284,283,283,279],[364,359,358,356,355,352,349,347,344,343,341,341,341,341,342,343],[411,423,430,434,438,453,457,462,471,475,480,484,496,500,506,511,514,514,512,510,505,503,498,497,494,493,493,494,496,498,502,508,510,515,522,527,532,534,537,538,538,535,532,529,522,517]],"t":[[0,6,11,23,27,27,39,43,56,60,72,77,89,93,106,110,118,127,130],[455,461,472,477,489,494,506,511,522,527,538,543,555,561,571,577,577,585,593,605,610,622,627,638,643],[912,923,930,935,939,944,955,961,972,977,988,994,1005,1010,1022,1027],[1275,1295,1302,1304,1310,1322,1327,1327,1339,1344,1344,1352,1360,1372,1377,1385,1394,1411,1419,1427,1438,1445,1455,1461,1469,1477,1489,1494,1502,1510,1519,1527,1539,1544,1555,1561,1572,1577,1589,1594,1595,1606,1610,1611,1622,1627]],"version":"2.0.0"}.
    • The gradient of a line is a measure of how steep it is (or sloppy it is). An alternative term for gradient is slope.
    • The gradient of a straight line given by m=tanθ=-ab{"x":[[105,104,104,103,102,102,101,102,103,105,108,109,111,112,114,117,120,124,125,132,143,149,152,158,160,165,167,170,171,173,173,173,173,173,172,172,176,179,184,189,196,204,208,221,225,235,238,241,247,249,251,253,257,258,260,260,260,260,259,258,257,256,255,255,255],[354,368,376,380,388,396,404,407,410,413,416,418,420],[343,349,365,370,384,388,401,405,416],[494,494,494,494,494,492,491,489,488,485],[471,479,487,491,496,506,519,527,531],[566,569,570,573,573,574,574,573,571,567,565,563,558,555,548,544,540,538,537,535,535,535,537,538,539,541,544,548,553,555,560,566,571,573,581,586,587,591,592,592,594,596,598,599,601,603],[622,622,622,621,621,622,623,624,625,625,625,625,625,625,624,624,623,623,623,622,622,622,622,621,621,621,622,624,627,630,634,639,643,646,652,654,658,660,662,662,663,663,664,664,665,666,666,667,668,669,670],[722,721,720,718,717,713,711,708,708,707,707,707,707,709,711,713,717,719,721,726,728,733,736,745,750,755,756,759,759,758,756,754,750,745,742,736,733,727,722,719,720,721,727,730,740,743,751],[808,817,820,823,830,835,841,846,849,853,855,857],[814,815,819,821,832,836,840,848,851,855,862,868],[932,933,935,936,940,945,953,956,965,968,971],[1065,1056,1051,1046,1044,1038,1035,1033,1028,1024,1022,1022,1024,1025,1030,1036,1039,1048,1051,1060,1066,1069,1074,1075,1077,1077,1078,1078,1079,1080,1081,1082],[1013,1022,1024,1027,1030,1037,1048,1056,1060,1063,1069,1072,1075,1077,1081,1083,1084,1086],[1031,1031,1031,1030,1030,1027,1025,1024,1023,1022,1021,1018,1018,1017,1017,1017,1018,1019,1021,1024,1028,1031,1034,1040,1042,1048,1049,1050,1053,1055,1055,1056,1056,1056,1055,1055,1053,1050,1048,1045,1040,1035,1032,1029,1024,1022,1017,1016,1015]],"y":[[230,230,231,231,239,244,263,275,287,296,321,322,322,320,315,309,300,290,285,267,249,244,243,246,247,256,260,274,279,293,300,307,311,312,314,312,303,295,285,275,263,252,247,234,231,227,227,227,228,230,232,234,245,249,259,265,270,282,288,294,305,309,317,320,324],[236,233,231,231,229,228,226,226,226,226,226,226,226],[285,282,279,278,277,277,275,275,274],[180,184,188,198,205,226,233,252,259,278],[243,231,226,224,221,216,209,206,205],[263,256,253,245,243,237,235,232,229,228,228,229,231,234,242,250,258,262,266,273,277,279,281,282,282,282,282,279,276,274,269,264,260,259,257,259,261,267,269,272,276,279,280,280,280,278],[241,240,239,237,236,235,235,234,235,236,238,239,242,244,249,250,257,259,262,267,270,272,277,279,283,285,286,285,282,278,272,266,260,257,250,250,250,253,256,258,261,265,267,269,270,274,275,276,277,278,278],[196,192,192,198,201,217,224,245,252,259,271,282,287,294,297,299,302,303,303,302,300,295,292,276,261,246,238,213,207,191,187,183,180,178,178,180,182,189,197,212,218,220,224,225,228,229,229],[231,230,230,230,230,230,230,230,230,230,230,231],[264,264,264,263,260,259,259,258,257,257,256,256],[230,230,230,229,229,229,228,228,226,226,226],[147,145,147,150,152,158,160,163,170,179,183,185,187,187,187,185,184,179,177,172,169,168,167,168,171,173,175,180,184,188,190,192],[250,249,249,249,249,248,246,245,245,244,244,244,244,244,244,244,244,246],[280,281,283,292,296,311,320,325,334,339,344,355,358,363,364,366,366,366,365,362,358,354,353,348,348,347,347,347,350,353,354,358,360,361,365,367,369,372,374,377,379,381,381,381,381,381,378,376,371]],"t":[[0,5,9,14,22,30,39,51,56,68,100,105,117,122,134,138,151,155,164,172,191,197,205,218,222,234,239,251,255,264,272,284,289,301,305,330,339,351,355,368,372,385,389,401,405,417,422,422,430,439,439,451,455,468,472,473,484,489,489,501,505,508,518,523,524],[820,839,847,850,856,864,872,881,884,889,889,901,903],[1106,1109,1114,1122,1133,1139,1147,1157,1167],[1383,1402,1409,1417,1422,1434,1439,1451,1457,1467],[1646,1665,1669,1674,1684,1689,1701,1706,1714],[1949,1958,1966,1973,1984,1989,1990,2001,2006,2017,2023,2025,2034,2039,2051,2056,2068,2073,2073,2084,2089,2090,2101,2106,2106,2114,2123,2134,2139,2140,2148,2157,2164,2173,2190,2198,2206,2217,2223,2223,2231,2239,2251,2256,2264,2273],[2370,2375,2383,2390,2401,2406,2418,2423,2440,2440,2451,2456,2468,2473,2485,2490,2501,2506,2507,2518,2523,2523,2535,2539,2551,2556,2557,2580,2584,2590,2601,2607,2618,2623,2631,2640,2651,2656,2668,2673,2673,2685,2689,2690,2702,2706,2707,2718,2723,2723,2735],[2905,2914,2917,2931,2940,2951,2956,2968,2973,2973,2985,2989,2998,3006,3007,3018,3023,3023,3035,3040,3040,3052,3056,3068,3073,3085,3090,3102,3106,3119,3123,3123,3135,3140,3152,3156,3161,3168,3173,3191,3198,3206,3218,3223,3231,3241,3248],[3514,3532,3540,3542,3552,3556,3569,3573,3585,3590,3590,3602],[3783,3810,3818,3823,3835,3840,3840,3852,3857,3857,3869,3873],[4088,4096,4099,4107,4117,4123,4135,4140,4152,4157,4157],[4439,4458,4466,4474,4486,4490,4491,4498,4507,4519,4524,4536,4540,4552,4557,4569,4574,4586,4590,4602,4607,4619,4624,4636,4640,4641,4649,4657,4665,4674,4685,4690],[4989,5007,5016,5019,5025,5036,5040,5052,5057,5057,5066,5074,5074,5086,5090,5091,5103,5107],[5305,5315,5324,5332,5342,5352,5357,5369,5374,5374,5382,5391,5402,5407,5408,5420,5424,5436,5441,5452,5457,5469,5474,5486,5491,5503,5507,5508,5520,5524,5536,5540,5541,5553,5557,5558,5570,5574,5575,5586,5591,5603,5607,5611,5619,5624,5636,5641,5649]],"version":"2.0.0"} where θ{"x":[[438,437,432,430,429,429,427,424,421,419,417,416,416,417,418,424,427,435,439,450,453,462,470,474,478,487,491,495,506,510,516,519,523,527,531,532,535,536,536,534,531,525,522,516,513,505,496,492,477,465,454,449,400,397,398,402,410,429,445,474,483,503,524]],"y":[[251,254,262,266,270,274,286,306,321,337,354,372,390,408,417,440,447,461,464,471,471,471,468,465,461,452,446,439,417,409,391,382,364,345,326,316,288,269,251,234,218,203,196,185,181,175,171,170,174,181,191,197,288,303,317,326,332,338,338,336,335,331,327]],"t":[[0,5,14,22,22,30,39,52,55,68,73,85,89,97,105,118,122,134,138,151,155,168,172,172,184,188,189,197,205,218,222,223,234,238,251,255,264,272,284,288,301,305,318,322,322,334,339,351,355,368,372,384,422,434,439,451,456,472,480,488,497,508,513]],"version":"2.0.0"} is the angle the line makes with the positive x-axis.
    • A straight line can be alternatively expressed in a slope-intercept form where we can deduce the slope and the y-intercept directly.
    Frequently Asked Questions about Gradient and Intercept

    What is Gradient and Intercept?

    Gradient is a measure of how steep a straight line is. The non-zero coordinate of the point where a straight line intersects either of the axes, is known as the Intercept of that straight line.

    What is the relationship between Gradient and Intercept?

    The gradient of any straight line is equal to the negative of the ratio between the y and x-intercepts.

    What is the formula for solving gradient and intercept?

    For a straight line given by ax+by+c=0, the x and y intercepts are given by -c/a and -c/b. The gradient of that line is given by -b/a.

    How do you graph gradient and intercept?

    Using the x and y-intercepts, one can plot the two points and then connect them in order to get the required line. And the gradient is simply the slope of that line.

    What is an example of gradient and intercept?

    Suppose a straight line is given by x-y+1=0, the y-intercept of this line will be at (0,1) and the x-intercept will be at (-1,0) and the gradient of this line will be 1.

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