Equation of the Line Through Two Points

Finds the slope a and the intercept b of the line y = ax + b through the two points (x₁, y₁) and (x₂, y₂).

Explanation

Two points determine exactly one line. This finds the slope $a$ and the intercept $b$ that describe it as $y = ax + b$ .

a = \dfrac{y_2 - y_1}{x_2 - x_1}, \quad b = y_1 - a x_1

$(x_1, y_1)$ , $(x_2, y_2)$ — the two points on the line
$a$ — the slope: how far $y$ climbs for every 1 that $x$ moves right
$b$ — the intercept: the $y$ coordinate where the line crosses the $y$ axis, which is the value of $y$ when $x = 0$

Slope is rise over run. Once you have it, substituting either point gives the intercept.

Example

Take the line through $(1, 2)$ and $(3, 8)$ .

a = \dfrac{8 - 2}{3 - 1} = \dfrac{6}{2} = 3

The intercept is $b = 2 - 3 \times 1 = -1$ , so the line is $y = 3x - 1$ . Checking against the other point, $3 \times 3 - 1 = 8$ , which is indeed $(3, 8)$ .

Notes

If $x_1 = x_2$ the denominator is zero and there is no slope. The line through those points is vertical; it cannot be written as $y = ax + b$ at all, only as $x = x_1$ . The calculator flags this.
A positive slope rises to the right, a negative one falls, and a slope of 0 is a horizontal line.
Two identical points determine nothing. Infinitely many lines pass through a single point.
The intercept also comes from $b = y_2 - a x_2$ . Either point gives the same value, which makes it a handy check.