Finding the equation of a line
We are often asked to find the equation of describing a linear relationship given certain information about the graph and table of values or the context.
To find the equation of a line in gradient-intercept form, we need to be able to identify the gradient and the $y$y-intercept of a line. Then substitute these values into the form:
$y=mx+c$y=mx+c, where $m$m is the gradient and $c$c is the $y$y-intercept.
From a graph
From a graph we will look for two easily identifiable points to find the equation. If it is clear on a graph one convenient point would be the $y$y-intercept. Then the gradient can be found using $m=\frac{rise}{run}$m=riserun.
From a table of values
A linear function is a relationship that has a constant rate of change. This means that the $y$y-values change by the same amount for constant changes in $x$x-values. We can calculate the gradient by analysing the change in $y$y-values with respect to the $x$x-values.
Worked example
Example 1
(a) Determine the gradient for the linear function represented by the following table of values:
| $x$x | $2$2 | $3$3 | $4$4 | $5$5 |
|---|---|---|---|---|
| $y$y | $9$9 | $6$6 | $3$3 | $0$0 |
Think: The $x$x-values increase by the same amount. We can determine how the $y$y-values change for a given change in $x$x in order to calculate the gradient.
Do:
We see a constant change in $y$y (the rise) of $-3$−3 for an equivalent change in $x$x (the run) of $1$1. Therefore, we can calculate the gradient as follows:
| $m$m | $=$= | $\frac{\text{rise}}{\text{run}}$riserun |
| $=$= | $\frac{-3}{1}$−31 | |
| $=$= | $-3$−3 |
(b) Determine the equation of the line, passing through the points given in the table, in gradient-intercept form.
Think: From part (a) we have that the relationship between $x$x and $y$y is of the form: $y=-3x+c$y=−3x+c. For the first value in the table, when $x=2$x=2, $y=9$y=9, what would we have to add to $-3\times2$−3×2 to obtain $9$9?
Do: $-6+15=9$−6+15=9, so $c=15$c=15. Hence, the linear rule that fits the table of values is $y=-3x+15$y=−3x+15.
Reflect: Check the rule matches the other sets of points in the table.
Practice questions
QUESTION 1
Write the equation of a line which has gradient $\frac{2}{3}$23 and goes through the point $\left(0,3\right)$(0,3).
Express the equation in gradient-intercept form.
question 2
Write an equation for $y$y in terms of $x$x.
| $x$x | $0$0 | $\ldots$… | $8$8 | $9$9 | $10$10 | $11$11 |
|---|---|---|---|---|---|---|
| $y$y | $-6$−6 | $\ldots$… | $18$18 | $21$21 | $24$24 | $27$27 |
Question 3
Consider the function in the graph.
What is the gradient?
What is the value of the $y$y-intercept?
Write the equation of the line in gradient-intercept form.