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=12
Hence when x=3 on the line, y=12. So the point (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=2−2=−1
Hence when y=8 on the line, x=−1. So the point (−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:
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 5.