Did you know real-life problems can be represented mathematically so the perfect solutions could be found for them?
For example, say an aircraft engineer needs an equation to demonstrate the elevation of a plane so that he can study the accuracy of its flight. Or a data analyst wants to determine a formula that predicts future expenditure and profitable income for a company. Or maybe you want to come up with a special recipe that calculates your savings over the year from all that pocket money you've kept.
What if I told you that all of this can be done? In this article, you will learn the various ways in which Equations can be written.
Defining Equations
We shall begin our discussion by defining the topic at hand.
Writing Equations is the process of writing a mathematical statement that contains equal signs.
To approach this, we can use mathematical symbols to express problems in a short and concise way such that the solutions could be found using mathematical processes.
Consider you have a phrase that says "a number \(x\) times 3 is equal to 120".
To find this number represented as \(x\) can be quite a hectic task to do mentally or by trial and error. However, when this is modelled into an equation, it becomes pretty simpler to tackle.
This phrase can be represented as
\[3x=120\]
To find a solution for this, we need to isolate the x by dividing each side by 3. Which will have us at 40. Therefore, the Number x here is 40.
Writing Equations Using Symbols
Writing equations using symbols involves representing complex statements with symbols such that they can be tackled with a more mathematical approach.
In this section, when you are presented with word problems, you need to state clearly what you want to represent each variable with. When you encounter problems like those, consider the following tips when solving them.
Familiarise yourself with the problem and understand it.
Convert the problem into an equation by identifying variables and indicating what they represent.
Let us go through an example that demonstrates this technique.
If Kelvin has three apples, and his brother Mike, on his way from school buys 5 more for him, how many apples does Kelvin have in all?
Solution
Carefully examining the problem, we realise the variable here is the unknown quantity that we are supposed to find. We can represent that with \(x\). If Kelvin has 3 already, and his brother adds 5 more, that makes it 3 + 5 then. The equation can then be modelled as
\[3+5=x\]
We can go ahead to solve this to see how many apples Kelvin must have now.
\[8=x\]
This also means;
\[x=8\]
Therefore, from the problem, Kelvin should have 8 apples now.
Here is another example for you!
The entrance fee to a monkey sanctuary cost $162 for 12 kids and 3 adults. At the same sanctuary, 8 kids and 3 adults also spent $122 on tickets. How much did each kid and adult have to pay?
Solution
Understanding the problem means we will have to break them down enough.
12 kids and 3 adults spend $162
8 kids and 3 adults spend $122
Let us now identify the variables in the equation.
Let \(x\) represent the cost of kids' tickets
Let \(y\) represent the cost of adults' tickets
Ticket cost for 12 kids + 3 adults is $162
Ticket cost for 8 kids + 3 adults is $122
\[12x+3y=162\]
\[8x+3y=122\]
Now let us see how well we can solve these mathematically. These are called systems of equations. They possess two variables (in this case, \(x\) and \(y\)) and require two equations to solve.
To find the values of the variables in an equation like this, one would either need to do it by substitution or by the elimination method. Let us use the elimination method here.
The elimination method involves adding or subtracting the equations such that one variable is eliminated from the equation. By this, the remaining variable can be found algebraically.
Now subtract the second equation from the first.
\[12x+3y-(8x+3y)=162-122\]
This becomes
\[4x=40\]
Next, simplify the equation obtained.
\[x=10\]
Now we can substitute the value of \(x\) into any of the equations to find \(y\). For this example, we will substitute it into the second equation.
\[8(10)+3y=122\implies 80+3y=122\]
Then
\[3y=122-80\implies 3y=42\]
and finally
\[y=14\]
Remember we let \(x\) represent kids' tickets, and \(y\) represent adult tickets? This means that the entrance fee costs $10 for kids and $14 for adults.
Writing Equations in Standard Form
In this segment, you will be introduced to writing equations in Standard Form. Before we begin, let us define what it means for an equation to be in standard form.
The Standard Form is a way of representing mathematical concepts such as equations in specific rules such that they appear in a common way.
The different forms of equations have different ways they are represented in standard form and this is going to be discussed below.
Linear Equations in Standard Form
Linear equations are such that they appear in a straight line when graphed. In its definition, it is mentioned that the highest exponent of the variable is not more than 1.
Therefore, the standard form of linear equations in one variable is represented as,
\[ax+b=10\]
where \(a\neq 0\) and \(x\) is a variable. Here is an example.
Two-variable linear equations in standard form are presented as;
\[ax+by+c=0\]
where \(a,b\neq 0\), with \(x\) and \(y\) being variables. Below is an example of this nature of linear equations.
Quadratic Equations in Standard Form
Quadratic Equations are such that they are equations of degree 2. This means that the highest exponent of their variable is 2. There are quite a Number of forms in which quadratic equations are represented. However, the standard form for quadratic equations is,
\[ax^2+bx+c=0\]
where \(a\neq 0\) and
\(a\) = coefficient of \(x^2\);
\(b\) = coefficient of \(x\);
\(c\) = constant
with \(a\), \(b\), and \(c\) being Real Numbers. Here is an example.
Writing Equations Based on a Table
Given a table, we are able to use its information to write an equation even without plotting the graph. Assuming we have a linear equation, what it really means is that there is a linear increment in the values of \(x\) projected to \(y\). That is why they are straight lines when plotted on a graph.
To find linear equations for example from a table, you need to find the slope of the line with its formula,
\[m=\frac{y_2-y_1}{x_2-x_1}\]
then we can now find the \(y\)-intercept algebraically. Let's look at an example.
Given the table below, where \(x\) values are mapped to the \(y\) respectively, write the equation in the slope-intercept form associated with the information.
\(x\) | 100 | 200 | 300 | 400 | 500 |
\(y\) | 14 | 20 | 26 | 32 | 38 |
Solution
Let us identify what the slope-intercept form of a line really is before we get to find the solution.
\[y=mx+b\]
where
\(y\) = \(y\) values in the coordinate plane
\(m\) = slope of the line
\(x\) = \(x\) values in the coordinate plane
\(b\) = \(y\)-intercept
First, we will find the slope of the line from the information. The formula for the slope of the line is;
\[m=\frac{y_2-y_1}{x_2-x_1}\]
This means if we take a value from the \(x\)-axis, we will take another corresponding value from the \(y\)-axis.
if \(y_2=20\), then \(x_2=200\);
if \(y_1=14\), then \(x_1=100\).
Then,
\[m=\frac{20-14}{200-100}=\frac{6}{100}=0.06\]
Since we have found the slope of the line, we can now substitute corresponding values of \(x\) and \(y\) into the equation including the slope so that we can find what the \(y\)-intercept is. Let us use the first values where;
\[x=100\]
\[y=14\]
Then by the slope-intercept form,
\[14=0.06(100)+b\implies 14=6+b\]
Solving for \(b\) yields
\[b=14-6=8\]
What we can do now is to substitute the \(y\)-intercept and the slope we have found into the equation in slope-intercept form. Therefore, the equation of the line here is,
\[y=0.06x+8\]
Writing the Equation of a Line
Writing the equation of a line is usually associated with finding the equation of a plotted line on a graph. By that, we will learn how to write linear equations from two given points.
The Slope-Intercept Form
The slope of a line explicitly projects the change in the \(y\) coordinate of a line with respect to the \(x\) coordinate. To write a linear equation in standard form, we isolate \(y\). The coefficient of \(x\) becomes the slope, and the constant then is the \(y\)-intercept. The stars form of the slope of a line is given as,
\[y=mx+b\]
where
\(y\) = \(y\) values in the coordinate plane
\(m\) = slope of the line
\(x\) = \(x\) values in the coordinate plane
\(b\) = \(y\)-intercept
Below is an example of an equation in this form.
Finding the Slope of a Line
The slope of a line is also known as the gradient. This speaks to how much the line is slant. A line can be absolutely horizontal and parallel to the \(x-\)axis if the slope is 0. However, if it is parallel to the \(y-\)axis, then it is considered undefined.
If we are given two coordinates of \((2, 8)\) and \((4, 3)\), the slope of the line is defined as
\[\frac{3-8}{4-2}.\]
This means that we are only subtracting the \(y\) component of the second point from the \(y\) component of the first point, whilst we subtract the \(x\) component of the second point from the \(x\) component of the first point. This is modelled in a formula as;
\[m=\frac{y_2-y_1}{x_2-x_1}.\]
In this case, you'll have
\[m=\frac{3-8}{4-2}.\]
By our example, we will have our slope as \(-2.5\).
Finding the y-intercept
Given the \(x-\) and \(y-\)values and finding the slope, now we have enough information to substitute this into the standard form equation to find the \(y-\)intercept. If one point is plugged into the equation, it should be able to give us the unknowns. Here we will use the first point; \((2, 8)\).
\[y=mx+b\implies 8=-2.5(2)+b\implies 8=-5+b\]
Now solving for \(b\) by making \(b\) the subject yields,
\[b=8+5=13\]
This means that the equation for this line is
\[y=3.5x+13\]
Here is another worked example.
Given the points are \((4, 3)\) and \((6, -2)\). Find the equation for the line.
Solution
Step 1: Find the slope of the line.
\[m=\frac{-2-3}{6-4}=-2.5\]
Step 2: Find the \(y\)-intercept.
Take the first point and substitute that into the standard form of linear equations
\[3=-2.5(4)+b\implies 3=-10+b\]
As before, solving for \(b\) gives us
\[b=3+10=13\]
Therefore, the linear equation here is
\[y=-2.5x+13\]
Writing the Equation of a Circle
In this final section, we will look at writing the equation for a circle. A circle is made from a continuous curve joined end to end. It does not have any straight sides or corners. A circle falls under the category of two-dimensional shapes. This means that we can construct its figure on a Cartesian coordinate plane comprising the \(x\) and \(y-\)axes. Before we begin, let us define a circle.
A circle is a type of conic section that is highly symmetric.
A circle is defined by the set of all points that are equidistant from a given point, called the centre.
There are two primary elements that make up a circle, namely, the centre and the radius. The diagram below illustrates this.
Fig. 1. Graphical representation of a circle.
Standard Form of a Circle
The standard form of a circle of radius \(r\) and centre \((h, k)\) is defined by the equation
\[(x-h)^2+(y-k)^2=r^2.\]
In some cases, we may need to find the centre of a given circle. In order to do this, we can use the midpoint formula. This is shown below.
The Midpoint Formula
Say we are provided with two coordinates of the endpoints of the diameter for a given circle. The Midpoint Formula for finding the centre \((h, k)\) of a circle is given by
\[(h, k)=(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}).\]
On the other hand, if we need to find the radius of a circle, we can use the distance formula. This is shown below.
The Distance Formula
Say we are provided with two coordinates of the endpoints of the diameter for a given circle. The Distance Formula for finding the radius \(r\) of a circle is given by
\[r=\sqrt{(y_2-y_1)^2+(x_2-x_1)^2}\]
Here is an example of an Equation of a circle.
Find the equation for a circle if the endpoints of one of its diameters are at \((1, –2)\) and \((3, 4)\).
Solution
Let \((x_1, y_1)=(1, -2)\) and \((x_2, y_2)=(3, 4)\)
We begin by evaluating the centre of the circle by the Midpoint Formula.
\[(h, k)=\left(\frac{1+3}{2}, \frac{-2+4}{2}\right)=\left(\frac{4}{2}, \frac{2}{2}\right)\]
Solving this yields
\[(h, k)=(2, 1)\]
We must now find the radius using the Distance Formula.
\[r=\sqrt{(4-(-2))^2+(3-1)^2}=\sqrt{6^2+2^2}=\sqrt{40}\]
Simplifying this yields,
\[r=2\sqrt{10}\]
Thus, \(r^2=40\). Now, plugging these values into the standard form of a circle, we obtain
\[(x-2)^2+(y-1)^2=40\]
Writing equations - Key takeaways
- Writing equations is the process of writing a mathematical statement that does contain equal signs.
- The standard form is a way of representing mathematical concepts such as equations in specific rules such that they appear in a common way.
- The standard form of linear equations in one variable is \(ax+b=0\).
- The standard form of linear equations in two variables is \(ax+by+c=0\).
- The standard form for a quadratic equation is \(ax^2+bx+c=0\)
- The standard form of the slope-intercept of a line is \(y=mx+b\).
- The process of finding the equation of a line from a plotted graph means first finding the slope of the line, and then finding the y-intercept.
- The standard form for a circle is \((x-h)^2+(y-k)^2=r^2\).
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