Intercept Form

How to solve intercept form problems (linear equations): formula, proof, examples, and their solutions.

Formula

The linear equation in intercept form is
x/a + y/b = 1.

a: x-intercept of the line
b: y-intercept of the line

x-intercept, y-intercept

Proof

The x-intercept is (a, 0).
And the y-intercept is (0, b).

The change of x is a - 0 = a.
And the change of y is 0 - b = -b.

Then the slope of the line is
m = -b/a.

Slope of a line

The slope of the line is -b/a.
And the y-intercept is b.

Then the linear equation in slope-intercept form
is y = [-b/a]x + b.

Divide both sides by b.

Move -x/a to the left side.

Then x/a + y/b = 1.

Example 1

The x-intercept is -3.
And the y-intercept is 2.

Then the linear equation in intercept form
is x/(-3) + y/2 = 1.

Change the linear equation
in slope-intercept form.

Move x/(-3) to the right side.

Multiply 2 on both sides.

Then y = (2/3)x + 2.

Example 2

To find the intercepts clearly,
change -y/4 to y/(-4):
x/5 + y/(-4) = 1.

The x-intercept is 5.
And the y-intercept is -4.

To graph the linear equation:

Point the x-intercept on the x-axis: (5, 0).
And point the y-intercept on the y-axis: (0, -4).

Draw a line that passes through these two intercepts.