Two-Point Form

Two-Point Form

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

Formula

Two-Point Form: y = [(y2 - y1)/(x2 - x1)](x - x1) + y1, y = [(y2 - y1)/(x2 - x1)](x - x2) + y2,. (x1, y1) and (x2, y2): Points on the line

(x1, y1) and (x2, y2) are the points on the line.

Then the slope of the line is [(y2 - y1)/(x2 - x1)].

Slope of a line

The slope is [(y2 - y1)/(x2 - x1)].
And the line passes through (x1, y1).

Then the linear equation in point-slope form
is y = ((y2 - y1)/(x2 - x1))⋅(x - x1) + y1.

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 (x2, y2)
to write the linear equation
in two-point form:
y = [(y2 - y1)/(x2 - x1)]⋅(x - x2) + y2.

Example

Use two-point form to write the linear equation of the given line. (2, 1) and (5, 4): The points on the given line

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

Use two-point form to write the linear equation of the given line. (2, 1) and (5, 4): The points on the given line

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.