Point-Slope Form

Point-Slope Form

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

Formula

Point-Slope Form: y = m(x - x1) + y1. m: Slope of the line, (x1, y1): Point on the line

The linear equation in point-slope form is
y = m(x - x1) + y1.

m: Slope of the line
(x1, y1): Point on the line

Slope of a line

Proof

Point-Slope Form: Proof of the Formula

Think of a slope between (x1, y1) and (x, y).
(x, y) is another point on the line.

The change of x is [x - x1].
And the change of y is [y - y1].

Then the slope of the line is (y - y1)/(x - x1).

Then set (y - y1)/(x - x1) = m.

Multiply [x - x1] on both sides.

Move -y1 to the right side.

Then y = m(x - x1) + y1.

Example 1

Use point-slope form to write the linear equation of the given line.

This line passes through (1, 3).

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

So the linear equation is y = 2(x - 1) + 3.

When writing a linear equation as an answer,
you should write it in slope-intercept form.

So change the right side
to make slope-intercept form.

Then y = 2x + 1.

Example 1: Another Solution

Use point-slope form to write the linear equation of the given line.

Let's solve the same example
by choosing another point, (-2, -3),
and see if you can get the same answer.

So this line passes through (-2, -3).

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

So the linear equation is y = 2(x - (-2)) - 3.

Change the linear equation
in slope-intercept form.

Then y = 2x + 1.

As you can see,
you can get the same answer.

Example 2

Graph the given linear equation on the coordinate plane. y = -(x + 1) + 2

To see (x1, y1) clearly,
change the given linear equation like this:
y = -(x - (-1)) + 2.

The slope is -1.

And the line passes through (-1, 2).

So start from (-1, 2).

The slope is -1.
So move 1 unit to the right
and move 1 unit downward.
Let's call this point the 'endpoint'.

Draw a line that passes through
(-1, 2) and the endpoint.

This line is the answer.