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Equation of a Line Through 2 Points

y = mx + c automatically, from (x₁,y₁) to (x₂,y₂).

(1, 3)  →  (2, 5)
Equation of the line:
y = 2x + 1
Gradient
m = 2
y-intercept
c = 1
Step-by-step solution
Step 1 — Find the gradient m:
m = (y₂ − y₁) / (x₂ − x₁) = (5 − 3) / (2 − 1) = 2 / 1 = 2
Step 2 — Find c by substituting the point (x₁, y₁):
y = mx + c → c = y₁ − m·x₁ = 3 − 2 · 1 = 1
Step 3 — Result:
y = 2x + 1
Other forms of the same line
Gradient-intercept form (slope-intercept)
y = 2x + 1
Standard form
2x − y = −1
Point-gradient form (point-slope)
y − 3 = 2·(x − 1)
Graph
-2-11234-2-112345670(1, 3)(2, 5)
Examples

How do you find the equation of a line through 2 points?

Every line in the plane has a unique equation, and two different points are enough to pin it down completely. The most common form is y = mx + c, where m is the gradient (also called the slope — how much y rises for every unit of x) and c is the y-value where the line crosses the y-axis.

y = mx + c — the standard gradient-intercept form of a line's equation. To find it from two points you do two steps: first work out the gradient m = (y₂ − y₁) / (x₂ − x₁), then substitute one of the points into the equation and solve for c: c = y − m·x.

Three worked examples

Example 1: the points (1, 3) and (2, 5).
m = (5 − 3) / (2 − 1) = 2 / 1 = 2; c = 3 − 2·1 = 1; y = 2x + 1.
Example 2: the points (0, 4) and (3, −2).
m = (−2 − 4) / (3 − 0) = −6 / 3 = −2; c = 4 − (−2)·0 = 4; y = −2x + 4.
Example 3: the points (−2, 5) and (2, −3).
m = (−3 − 5) / (2 − (−2)) = −8 / 4 = −2; c = 5 − (−2)·(−2) = 5 − 4 = 1; y = −2x + 1.

Common mistake

Guessing c incorrectly — many students try to read off c just by eye. Always substitute one of the points after you've found m: once you know the gradient, take (x₁, y₁) or (x₂, y₂) and substitute it into y = mx + c. Both points must give the same c — a great way to check you haven't made a mistake.

💼 Where it's used: physics (uniform motion — position as a function of time), economics (price-quantity graphs, supply and demand), unit conversion (Celsius ⇄ Fahrenheit), and any situation where two quantities are in a linear relationship. This is the most fundamental equation in linear algebra — without it you can't understand regression, graphs, or higher-order functions.

How to use it

  • Enter the two coordinates (x₁, y₁) and (x₂, y₂).
  • The tool works out m = (y₂−y₁)/(x₂−x₁) and then c = y₁ − m·x₁.
  • When x₁ = x₂ the line is vertical, so the gradient is undefined — it shows x = c.
  • Forms shown: gradient-intercept, standard, and point-gradient.

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✍️ Written by the MathsUK teamChecked against the National Curriculum and GCSE specifications