# Point-Slope Form

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

## Formula

The linear equation in point-slope form is*y* = *m*(*x* - *x*_{1}) + *y*_{1}.*m*: Slope of the line

(*x*_{1}, *y*_{1}): Point on the line

Slope of a line

## Proof

Think of a slope between (*x*_{1}, *y*_{1}) and (*x*, *y*).

(*x*, *y*) is another point on the line.

The change of *x* is [*x* - *x*_{1}].

And the change of *y* is [*y* - *y*_{1}].

Then the slope of the line is (*y* - *y*_{1})/(*x* - *x*_{1}).

Then set (*y* - *y*_{1})/(*x* - *x*_{1}) = *m*.

Multiply [*x* - *x*_{1}] on both sides.

Move -*y*_{1} to the right side.

Then *y* = *m*(*x* - *x*_{1}) + *y*_{1}.

## Example 1

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* = 2*x* + 1.

## Example 1: Another Solution

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* = 2*x* + 1.

As you can see,

you can get the same answer.

## Example 2

To see (*x*_{1}, *y*_{1}) 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.