How to Write the Equation of a Line from Two Points

Quick answer

To find the equation of a line through two points, first compute the slope with the slope formula, then substitute that slope and either point into point-slope form, then simplify. Through (1, 3) and (3, 7) the slope is 2, giving y - 3 = 2(x - 1), which simplifies to y = 2x + 1.

What you'll learn

Find the equation of a line through two given points
Combine the slope formula with point-slope form
Handle the special cases of horizontal and vertical lines

The whole method in one line

Two points determine a line, so they must determine its equation. Getting there is two tools you already have, used in order:

\text{two points} \;\xrightarrow{\text{ slope formula }}\; m \;\xrightarrow{\text{ point-slope }}\; \text{equation}

Compute the slope: $m = \tfrac{y_2 - y_1}{x_2 - x_1}$ .
Substitute $m$ and either point into $y - y_1 = m(x - x_1)$ .
Simplify to slope-intercept form if you want the intercept.

Why you need the slope first

Point-slope form needs a slope, and you are not given one — you are given two points. So the slope formula comes first, and its answer feeds the second step.

It is worth seeing that this always works. Two distinct points with different $x$ -values give exactly one slope, and a slope plus a point gives exactly one line. So the answer exists and is unique — there is never a choice to agonise over, only arithmetic to do carefully.

Worked examples

Common mistakes

Practice problems

Find the equation of the line through $(0, 2)$ and $(4, 10)$ .

Hint

Compute the slope first. One of these points is already the $y$ -intercept.

Answer

$y = 2x + 2$

Full solution

$m = \tfrac{10 - 2}{4 - 0} = \tfrac{8}{4} = 2$ .

Since $(0, 2)$ is the $y$ -intercept, $b = 2$ and $y = 2x + 2$ directly.

Check at $x = 4$ : $y = 10$ ✓
Find the equation of the line through $(1, 5)$ and $(3, 11)$ .

Answer

$y = 3x + 2$

Full solution

$m = \tfrac{11 - 5}{3 - 1} = 3$ . Using $(1, 5)$ : $y - 5 = 3(x - 1) = 3x - 3$ , so $y = 3x + 2$ .

Check at $x = 3$ : $y = 11$ ✓
Find the equation of the line through $(-1, 4)$ and $(2, -2)$ .

Answer

$y = -2x + 2$

Full solution

$m = \tfrac{-2 - 4}{2 - (-1)} = \tfrac{-6}{3} = -2$ .

Using $(2, -2)$ : $y + 2 = -2(x - 2) = -2x + 4$ , so $y = -2x + 2$ .

Check at $x = -1$ : $y = 2 + 2 = 4$ ✓
Find the equation of the line through $(3, -1)$ and $(8, -1)$ .

Hint

Compute the slope before reaching for point-slope form.

Answer

$y = -1$

Full solution

$m = \tfrac{-1 - (-1)}{8 - 3} = \tfrac{0}{5} = 0$ , so the line is horizontal at the shared height: $y = -1$ .
Find the equation of the line through $(-2, 3)$ and $(-2, 9)$ .

Answer

$x = -2$

Full solution

The $x$ -values match, so the denominator is $0$ and the slope is undefined. The line is vertical: $x = -2$ .
Find the equation of the line through $(0, -3)$ and $(6, 1)$ .

Hint

Expect a fractional slope. Keep it as a fraction.

Answer

$y = \tfrac{2}{3}x - 3$

Full solution

$m = \tfrac{1 - (-3)}{6 - 0} = \tfrac{4}{6} = \tfrac{2}{3}$ .

$(0, -3)$ is the intercept, so $y = \tfrac{2}{3}x - 3$ .

Check at $x = 6$ : $y = 4 - 3 = 1$ ✓
Two students find the line through $(2, 1)$ and $(5, 7)$ . One writes $y - 1 = 2(x - 2)$ , the other writes $y - 7 = 2(x - 5)$ . Who is right?

Hint

Simplify both.

Answer

Both.

Full solution

First: $y - 1 = 2x - 4$ , so $y = 2x - 3$ .

Second: $y - 7 = 2x - 10$ , so $y = 2x - 3$ .

They used different points, which is allowed, and reached the same line.
A plant is measured at $12$ cm on day $3$ and $27$ cm on day $8$ . Assuming steady growth, write an equation for its height and predict its height on day $12$ .

Hint

Days are $x$ , height is $y$ . Find the growth rate first.

Answer

$y = 3x + 3$ ; on day $12$ it is $39$ cm.

Full solution

$m = \tfrac{27 - 12}{8 - 3} = \tfrac{15}{5} = 3$ cm per day.

Using $(3, 12)$ : $y - 12 = 3(x - 3) = 3x - 9$ , so $y = 3x + 3$ .

On day $12$ : $y = 36 + 3 = 39$ cm.

The intercept $3$ predicts a height of $3$ cm on day $0$ , which is a reasonable seedling — but predicting far outside the measured range assumes growth stays steady, and real plants eventually stop.

Frequently asked questions

Which of the two points should I substitute?

Either. Both give equations that simplify to the same line. Pick the one with smaller or positive numbers, since it makes the arithmetic cleaner.

What if the two points give a slope of zero?

Then the line is horizontal and its equation is y = the shared y-value. There is no need for point-slope form; the answer is already in front of you.

What if the x-values are the same?

Then the slope is undefined and the line is vertical. Its equation is x = the shared x-value. A vertical line cannot be written as y = mx + b at all.

What to learn next

Parallel and Perpendicular Lines: The Slope RulesParallel lines have equal slopes and perpendicular lines have negative reciprocal slopes — why both rules are true, and how to write equations using them.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

CCSS.MATH.CONTENT.HSA.CED.A.2Creating EquationsCreate equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
CCSS.MATH.CONTENT.8.EE.B.6Expressions and EquationsUse similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

For teachersPrint problems onlyTick to print a blank worksheet; leave it clear to print the lesson with the full answer key.