Slope-Intercept Form

Slope-Intercept Form

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

Formula

Slope-Intercept Form: y = mx + b. m: Slope of the line, b: y-intercept of the line

The linear equation in slope-intercept form is
y = mx + b.

m: Slope of the line
b: y-intercept of the line

Slope of a line

x-intercept, y-intercept

Proof

Slope-Intercept Form: Proof of the Formula

Think of a slope between (0, b) and (x, y).
(x, y) is the point on the line.

The change of x is, x - 0, [x].
And the change of y is [y - b].

Then the slope of the line is (y - b)/x.

Then set (y - b)/x = m.

Multiply x on both sides.

Move -b to the right side.

Then y = mx + b.

Example 1

Write the linear equation of the given line.

The y-intercept is -1.

And the slope, m, is 2/1 = 2.

So the linear equation is y = 2x - 1.

Example 2

Graph the given linear equation on the coordinate plane. y = 3x + 1

The slope is 3.

And the y-intercept is -1.

So start from the y-intercept: -1.

The slope is 3.
So move 1 unit to the right
and move 3 units upward.
Let's call this point the 'endpoint'.

Draw a line that passes through
the y-intercept and the endpoint.

This line is the answer.

Example 3

Write the linear equation 5x - 3y = 6 in slope-intercept form.

Slope-intercept form is y = mx + b.

There's only y term on the left side.

So move the x term, 5x, to the right side.

Divide both sides by -3.

Then y = (5/3)x - 2.

This linear equation is in slope-intercept form.
So this is the answer.

Let's graph y = (5/3)x - 2
on the coordinate plane.

Start from the y-intercept: -2.

The slope is 5/3.
So move 3 unit to the right
and move 5 units upward.
Let's call this point the 'endpoint'.

Draw a line that passes through
the y-intercept and the endpoint.