Last updated: 2026-05-19

The equation of a line

A straight line in the coordinate plane is completely determined by any two of its points. Given points P₁ = (x₁, y₁) and P₂ = (x₂, y₂), the line through them has a single slope and can be written in three standard forms — each useful in different contexts.

The three standard forms

Slope-intercept form: y = mx + b

The most commonly used form. Here m is the slope (rise ÷ run) and b is the y-intercept (where the line crosses the y-axis).

$m = \frac{y_2 - y_1}{x_2 - x_1}$

$b = y_1 - m \cdot x_1$

Best for: graphing (start at (0, b), then rise m for every 1 step right), reading off the slope at a glance, and functions-of-x problems.

Point-slope form: y − y₁ = m(x − x₁)

Uses the slope and one of the given points directly, without first computing the y-intercept.

Best for: writing the equation when you know a point and the slope, or when deriving other forms — it's the most direct statement of what a slope means.

Standard form: Ax + By = C

All three variables on one side, no fractions (when the input coordinates are integers). Convention: A ≥ 0.

Best for: linear systems (both equations in Ax + By = C form are easy to add or subtract), finding intercepts (set x = 0 → y = C/B; set y = 0 → x = C/A).

Special cases

Vertical lines

When x₁ = x₂, the two points share the same x-coordinate and lie on a vertical line. The slope is undefined (division by zero), so slope-intercept and point-slope forms don't apply. The equation is simply x = x₁.

Horizontal lines

When y₁ = y₂, the line is horizontal. The slope is 0, the y-intercept is y₁, and all three forms simplify to y = y₁.

Worked example

Given P₁ = (1, 2) and P₂ = (4, 11):

$m = \frac{11 - 2}{4 - 1} = \frac{9}{3} = 3$

$b = 2 - 3 \cdot 1 = -1$

• Slope-intercept: y = 3x − 1
• Point-slope: y − 2 = 3(x − 1)
• Standard: 3x − y = 1

Converting between forms

Integer coefficients in standard form

When the slope m = p/q is already a fraction (with p and q coprime integers), multiply every term of y = mx + b through by q:

$qy = px + qb \implies px - qy = -qb$

Now A = p, B = −q, C = −qb are all integers. This calculator applies this step automatically for integer input coordinates.