Finding the equation of a line
There are a range of methods that can be used to find the equation of a line given certain information about the graph, or the context. Depending on the information given, a particular form of a line might be more convenient. Here are some useful forms:
- $y=mx+c$y=mx+c (gradient-intercept form)
- $y-y_1=m\left(x-x_1\right)$y−y1=m(x−x1) (point-gradient formula)
- $\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}$y−y1x−x1=y2−y1x2−x1 (two point formula)
- $ay+bx-c=0$ay+bx−c=0 (general form)
Let's look at examples where the different forms may be convenient.
Gradient-intercept form
To use the gradient-intercept rule, two pieces of information are required: the gradient and the $y$y-intercept of the line. Once these are known, substitute the values into the form:
$y=mx+c$y=mx+c
where $m$m is the gradient and $c$c is the $y$y-intercept
If the gradient or $y$y-intercept are not known, such as when the gradient is provided with a different point, this method can still be used. This is done by substituting the gradient and known point to find the unknown $y$y-intercept. Alternatively, to achieve this in one step, use the point-gradient formula.
Practice question
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.
If this information is given in context, remember that the gradient represents the rate of change. That is, for each increase in the $x$x-value by $1$1, there is a change in the $y$y-value by $m$m. The $y$y-intercept is the value of $y$y when $x$x is zero. In context, this is often the initial value.
Practice question
QUESTION 2
A carpenter charges a callout fee of $\$150$$150 plus $\$45$$45 per hour.
Write an equation to represent the total amount charged, $y$y, by the carpenter as a function of the number of hours worked, $x$x.
What is the gradient of the function?
What does this gradient represent?
The total amount charged increases by $\$45$$45 for each additional hour of work.
AThe minimum amount charged by the carpenter.
BThe total amount charged increases by $\$1$$1 for each additional $45$45 hours of work.
CThe total amount charged for $0$0 hours of work.
DWhat is the value of the $y$y-intercept?
What does this $y$y-intercept represent?
Select all that apply.
The total amount charged increases by $\$150$$150 for each additional hour of work.
AThe maximum amount charged by the carpenter.
BThe callout fee.
CThe minimum amount charged by the carpenter.
DFind the total amount charged by the carpenter for $6$6 hours of work.
Point-gradient formula
If the $y$y-intercept is not known but the gradient and some other point on the line is known, use the point-gradient formula:
$y-y_1=m\left(x-x_1\right)$y−y1=m(x−x1)
Practice question
Question 3
A line passes through the point $A$A$\left(-4,3\right)$(−4,3) and has a gradient of $-9$−9. Using the point-gradient formula, express the equation of the line in gradient intercept form.
The two-point formula
If given two points, $\left(x_1,y_1\right)$(x1,y1) and $\left(x_2,y_2\right)$(x2,y2), use the two-point formula to find the equation of a line. This equation is actually similar to the point-gradient formula but requires calculating the gradient on the right-hand side using $m=\frac{rise}{run}$m=riserun since $m$m is not known. To find the equation, substitute the known coordinates into the formula:
$\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}$y−y1x−x1=y2−y1x2−x1
Practice question
Question 4
Find the equation of the line which passes through Point A $\left(8,1\right)$(8,1) and Point B $\left(-6,-2\right)$(−6,−2).
Use the two point formula and give your answer in gradient-intercept form.
General form
Another way of writing the equation of a straight line is called general form. A straight line is in general form when it is written with all terms on one side of the equation and zero on the other, and all coefficients are integers. In particular, the coefficient of $x$x should be positive (the sign can be easily changed by multiplying the whole equation by $-1$−1).
That is, a straight line is in general form when it is of the form
$ax+by+c=0$ax+by+c=0
for integers $a$a, $b$b and $c$c with $a>0$a>0. This form may be used to find the equation of a line parallel to another line given in this form.
Practice question
Question 5
Use the two point formula to derive the equation of the line that passes through the points $A$A$\left(6,5\right)$(6,5) and $B$B$\left(-16,11\right)$(−16,11). Give your answer in general form.
From a graph or table
From a graph or table, look for two easily identifiable points and use one of the methods above to find the equation. If it is in the table or can be read from the graph, one convenient point to use would be the $y$y-intercept. Then the gradient can be found using $m=\frac{rise}{run}$m=riserun.
Practice questions
Question 6
Consider the graph of the line.
What is the value of the $y$y-intercept?
What is the slope of the line?
Find the equation of the line in the form $y=mx+c$y=mx+c.
Rewrite the equation of the line in general form $ax+by+c=0$ax+by+c=0.
Question 7
After Mae starts running, her heartbeat increases at a constant rate.
Complete the table.
Number of minutes passed ($x$x) $0$0 $2$2 $4$4 $6$6 $8$8 $10$10 $12$12 Heart rate ($y$y) $49$49 $55$55 $61$61 $67$67 $73$73 $79$79 $\editable{}$ What is the constant rate the heart beat is increasing at?
Which one of the following equations describes the relationship between the number of minutes passed ($x$x) and Mae’s heartbeat ($y$y)?
$y=49x-3$y=49x−3
A$y=49x+3$y=49x+3
B$y=3x-49$y=3x−49
C$y=3x+49$y=3x+49
DIn the equation, $y=3x+49$y=3x+49, what does $3$3 represent?
The change in one minute of Mae’s heartbeat.
AThe total time Mae has run.
BThe total distance Mae has run.
C