# Two-Point Form

How to solve point-slope form problems (linear equations): formula, example, and its solutions (2 ways).

## Formula

(*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) are the points on the line.

Then the slope of the line is [(*y*_{2} - *y*_{1})/(*x*_{2} - *x*_{1})].

Slope of a line

The slope is [(*y*_{2} - *y*_{1})/(*x*_{2} - *x*_{1})].

And the line passes through (*x*_{1}, *y*_{1}).

Then the linear equation in point-slope form

is *y* = ((*y*_{2} - *y*_{1})/(*x*_{2} - *x*_{1}))⋅(*x* - *x*_{1}) + *y*_{1}.

This is two-point form of the linear equation.

As you can see,

two-point form

is just another version of point-slope form.

You can also use (*x*_{2}, *y*_{2})

to write the linear equation

in two-point form:*y* = [(*y*_{2} - *y*_{1})/(*x*_{2} - *x*_{1})]⋅(*x* - *x*_{2}) + *y*_{2}.

## Example

Instead of directly writing the linear equation

in two-point form,

first find the slope,

then use the point-slope form,

because it's easier to know what you're doing.

The change of *x* is 5 - 2 = 3.

And the change of *y* is 4 - 1 = 3.

Then *m* = 3/3

= 1.

Slope of a line

The slope is 1.

And the line passes through (2, 1).

Then the linear equation is*y* = 1(*x* - 2) + 1.

point-slope form

Change the linear equation

in slope-intercept form.

Then *y* = *x* - 1.

## Example: Another Solution

Let's choose the other point (5, 4)

and see if you can get the same answer.

The change of *x* is 5 - 2 = 3.

And the change of *y* is 4 - 1 = 3.

Then *m* = 3/3

= 1.

Slope of a line

The slope is 1.

And the line passes through (5, 4).

Then the linear equation is*y* = 1(*x* - 5) + 4.

point-slope form

Change the linear equation

in slope-intercept form.

Then *y* = *x* - 1.

As you can see,

you can get the same answer.