Find the gradient of a line through two points and determine the equation of a straight line in the form y = mx + c

What this dot point is asking

SEAB wants you to find the gradient of a line joining two points and to work out the full equation of a straight line, written as $y = mx + c$, from either two points or a point and a gradient. This builds directly on linear graphs and is the core skill of coordinate geometry.

The answer

The gradient between two points

The gradient measures steepness: how much $y$ changes for each unit of $x$. For two points $(x_1, y_1)$ and $(x_2, y_2)$:

Keep the same order top and bottom. For $A(2, 1)$ and $B(5, 7)$, the gradient is $\dfrac{7 - 1}{5 - 2} = \dfrac{6}{3} = 2$.

The equation of a straight line

Every non-vertical straight line has the form:

where $m$ is the gradient and $c$ is the $y$-intercept. To find the equation you need the value of both $m$ and $c$.

Finding the equation from two points

1. Find the gradient $m$ from the two points.
2. Write $y = mx + c$ with the known gradient.
3. Substitute one of the points to find $c$.
4. Write the complete equation.

Finding the equation from a point and a gradient

If the gradient is already known, skip straight to writing $y = mx + c$, then substitute the given point to find $c$. This is quicker because step 1 is done for you.

Horizontal and vertical lines

Two special cases are worth knowing. A horizontal line has gradient $0$, so its equation is simply $y = c$ (for example $y = 4$). A vertical line has an undefined gradient and is written as $x = a$ (for example $x = 3$); it cannot be put in the form $y = mx + c$ because $x$ never changes along it.

Checking the equation

Substitute the other point (or the given point) back into your equation. If both sides match, the line really does pass through that point and the equation is correct. This single check catches most arithmetic and sign slips, so it is always worth the few seconds it takes.

Examples in context

Example 1. A line of best fit. In statistics, a scatter graph's line of best fit is a straight line whose equation can be found by reading off two clear points on it and using $y = mx + c$. The gradient then describes the rate of change in the data, linking coordinate geometry to statistics.

Example 2. Parallel lines. Two lines are parallel when they share the same gradient. So the line through $(0, 1)$ parallel to $y = 3x - 4$ also has gradient $3$, giving $y = 3x + 1$ once the new intercept is found from the point. Recognising equal gradients saves recalculating the slope.

Try this

• Cue. Find the gradient through $(1, 1)$ and $(4, 7)$. Compute $\dfrac{7 - 1}{4 - 1} = 2$.
• Cue. A line has gradient $4$ through $(2, 5)$. Then $5 = 4(2) + c$, so $c = -3$ and $y = 4x - 3$.
• Cue. Find the equation through $(0, 6)$ and $(2, 2)$. Gradient $\dfrac{2 - 6}{2 - 0} = -2$, intercept $6$, so $y = -2x + 6$.