Graph a Line by Intercepts
Note that the y-intercept is also called vertical intercept, and x-intercept is also called horizontal
intercept. In this document, we will use y-intercept and x-intercept.
Earlier, when we graphed the line 2 x + 3 y = 24 , we first changed it to slope-intercept form
y=−
2
x + 8 , and then graphed it using its y-intercept and slope triangles.
3
Let's learn a second way to graph this line.
[Example 1] Graph 2 x + 3 y = 24 .
[Solution] To graph a line, we need to identify at least two points on the line.
To identify a point, we plug a value into x, and then solve for y. The easiest number in the world is 0, so
we plug x=0 into 2 x + 3 y = 24 , and we have:
2 x + 3 y = 24
2 ⋅ 0 + 3 y = 24
3 y = 24
3 y 24
=
3
3
y =8
The first point we found is (0, 8), which is the y-intercept.
Next, we could plug in x=1, but 1 is not the easiest number. When a line is given in standard form,
plugging in y=0 is easier:
2 x + 3 y = 24
2 x + 3 ⋅ 0 = 24
2 x = 24
2 x 24
=
2
2
x = 12
The second point we found is (12, 0), which is the x-intercept.
Now we can graph this line by graphing its x-intercept and y-intercept (thus the name Intercept Method):
Figure 1: Graph of 2x+3y=24
In some situations, this method is easier than changing from Ax + By = C to y = Mx + B . However,
sometimes we have to resort to the other method, as in Example 2.
[Example 2] Graph the line 2 x + 3 y = 0 .
[Solution] Let's try the intercept method as usual. First, plug in x=0, we have:
2x + 3y = 0
2 ⋅ 0 + 3y = 0
3y = 0
3y 0
=
3 3
y=0
The y-intercept is (0, 0). Next, plug in y=0, we have:
2x + 3y = 0
2x + 3 ⋅ 0 = 0
2x = 0
2x 0
=
2
2
x=0
So the x-intercept is (0, 0). Oops!
If a line passes the origin (0, 0), of course it has the same x-intercept and y-intercept. With only one
point, there is no way to graph the line!
What if we plug in x=1? Let's try it:
2x + 3y = 0
2 ⋅1 + 3y = 0
2 + 3y = 0
2 + 3y − 2 = 0 − 2
3 y = −2
3y − 2
=
3
3
2
y=−
3
2
3
The point we got is (1,− ) . It's difficult to graph a point with fractional coordinates!
Instead of trying to plug different values into x, it's easier to change the equation to slope-intercept form:
2x + 3y = 0
2x + 3y − 2x = 0 − 2x
3 y = −2 x
3y − 2x
=
3
3
2
y=− x
3
Now we can graph this line by its y-intercept (0, 0) and a slope triangle:
Figure 2: Graph of 2x+3y=0
Rule of thumb: When you graph a line given in standard form Ax + By = C , try the intercept method
first. If you end up with fractional coordinates, or if both intercepts turn out to be (0, 0), we need to
change the line to slope-intercept form y = Mx + B , and then use the slope-triangle method to graph
it.