Writing Linear Equations

Linear equations are equations for which the highest power of the variable is 1. It possesses x and y values in a way that they appear in a straight line when graphed on a Cartesian plane. Solving for linear equations means finding the values of the variables present in the equation such that when they are substituted they will make the equation true.

Linear equations can have either one variable, two variables, or three variables. Examples of one-variable linear equation is as follows;

1. $x+21=15$
2. $3y-4=y$
3. $6+2x+x=3$

Examples of two-variable linear equations are as follows;

1. $2x+5y=15$
2. $23-3x=4y$
3. $1=4x-23y$

Examples of three-variable linear equations are as follows;

1. $x+2y=z-4$
2. $4x-16y=2z+18$
3. $15x-4x+12=z-3y$

In what forms are linear equations written?

There are three forms in which linear equations are written, and they are;

• Standard form
• Slope intercept form
• Point slope form

Standard form of linear equations

Linear equations with one variable in standard form are presented as;

$ax+b=0$

Where $a\ne 0$

$x$ is a variable

Two-variable linear equations in standard form are presented as;

$ax+by+c=0$

Where $a\ne 0$

$b\ne 0$

$x$ and$y$ are variables

Three-variable linear equations in standard form are presented as;

$ax+by+cz+d=0$

Where $a\ne 0$

$b\ne 0$

$c\ne 0$

$x,y$ and $z$ are variables.

Let us look at an example of how two-variable linear equations will look like below;

Slope intercept form of linear equations

The slope-intercept form is probably the most common way you would come across linear equations. It is written in the form;

$y=mx+b$

Where $y=ycomponentonthegraph$

$m=slope$

$x=xcomponentonthegraph$

$b=y-intercept$

Point slope form of linear equations

A straight line is formed with regards to the coordinate plane in this form of writing linear equations. It is written in the form;

$y–y_1=m(x–x_1)$

Where $(x_1,y_1)$ are coordinates on the plane.

Function form of linear equations

In this form of writing linear equations, it is written as a function such that

$f\left(x\right)=x+C$

Here, $y$ is replaced with $f\left(x\right)$.

How to write linear equations with two points

Most problems associated with linear problems often appear to be out of you plotting the graph from a linear equation, where maybe, variables are supposed to be solved for. Here, it is rather going to be the other way around, where the equation is derived from the graph. By that, we will learn how to write linear equations from two given points, first by finding the slope of the line, then by finding the y-intercept.

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, whist we subtract the x component of the second point from the x component of the first point. This is modelled in a formula as;

$m(slopeofline)=\frac{y_2-y_1}{x_2-x_1}$

$m=\frac{3-8}{4-2}$

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).

$y=mx+b$

$$\begin{gathered} 8=-2.5\left(2\right)+b \\ 8=-5+b \\ 8+5=b \\ b=13 \end{gathered}$$

This means that the equation for this line is $y=-2.5x+13$

Writing linear equations from word problems

There are some word problems that will require being solved with linear systems. When such problems are encountered the following are tips to consider when solving them.

1. Familiarise yourself with the problem and understand it
2. Convert the problem into an equation by identifying variables and indicating what they present

We can look at an example that involves two variables.

Writing the linear equation of parallel lines