Equation of a straight line: $y=mx+c$

In a nutshell

The equation $y=mx+c$ gives a straight line on a coordinate grid, where $m$ and $c$ are constants. It is the equation for almost any straight line, the exception being vertical lines, which have equations of the form $x=d$ where $d$ is a constant (the $x$ -intercept). You can use a straight line graph to work out the equation of the line.

The components of the equation $y=mx+c$

$m$ is the value of the gradient of the straight line and $c$ is the $y$ -intercept. $x$ and $y$ correspond to coordinates of points on the line. For any point $(x,y)$ on the line, multiplying the $x$ -coordinate by $m$ and adding $c$ , gives the $y$ -coordinate. If this doesn't work, then the point you are using is not actually on the line.

Finding points on a line

If you have a straight line graph $y=mx+c$ then you will be given $m$ and $c$ to specify the line. Concurrently, $x$ and $y$ are not given as specific values, since they represent the $x$ - and $y$ -coordinates of any point on the line. If you insert an $x$ -coordinate into the equation of the line, it gives the corresponding $y$ -coordinate for the point on the line.

Example 1

Take the line with equation

$y=7x-9$

What are the coordinates of the point on this line that has $x$ -coordinate $3$ ?

Substitute $x=3$ into this equation:

$y=7x-9=7(3)-9=21-9=\underline{12}$

Hence when $x=3$ on the line, $y=12$ . So the point $\underline{(3,12)}$ is on the line with equation $y=7x-9$ .

Similarly, you can find a corresponding $x$ -coordinate if given a $y$ -coordinate of a point on a line. It just requires some algebraic rearranging to make $x$ the subject of the equation of the line.

Example 2

Take the line with equation

$y=-2x+6$

What are the coordinates of the point on this line that has $y$ -coordinate $8$ ?

Start by substituting $y=8$ into the equation:

$8=-2x+6$ 

Now rearrange to make $x$ the subject:

$8+2x=6$

$2x=6-8=-2$

$x=\frac{-2}2=\underline{-1}$

Hence when $y=8$ on the line, $x=-1$ . So the point $\underline{(-1,8)}$ is on the line with equation $y=-2x+6$ .

Using a graph to solve a linear equation

If you have the graph of a linear equation, you can solve a corresponding linear equation for a given $x$ - or $y$ -value. Simply read off the corresponding value from the graph.

Example 3

Consider the graph of $y=2x-4$ :

Maths; Graphs; KS4 Year 10; Equation of a straight line: y = mx + c

Using the graph, solve the equation

$6=2x-4$ 

The equation you have to solve is the equation of the line but with $y=6$ . Hence read off the graph to find the $x$ -coordinate when $y=6$ . You find that the $x$ -coordinate is $\underline{5}$ .