Write the equation of a line given points, slope, or a graph. | Linear equations in two variables | SAT Math

Find a slope and a point, then use point-slope form to write the equation.

Core Idea

To write a line's equation, you need a slope and a point. Two points give you the slope via $𝑚 = 𝑦 2 - 𝑦 1 𝑥 2 - 𝑥 1$ , and then point-slope form builds the equation.

Understanding

The fastest path to an equation depends on what you're given. If you have the slope and a point, plug straight into point-slope form $𝑦 - 𝑦 1 = 𝑚 (𝑥 - 𝑥 1)$ and simplify. If you have two points, compute the slope first, then use either point.

From a graph, pick two clear lattice points (where the line crosses grid intersections), calculate the slope from rise over run, and read the y-intercept if the line crosses the y-axis at a grid point.

The most common mistake is mixing up the order of subtraction in the slope formula — just be consistent: $𝑦 2 - 𝑦 1$ on top, $𝑥 2 - 𝑥 1$ on the bottom, using the same point as "point 2" in both.

Step by Step

If given two points, compute slope: $𝑚 = 𝑦 2 - 𝑦 1 𝑥 2 - 𝑥 1$ .
Pick one of the points (or the given point) and substitute into $𝑦 - 𝑦 1 = 𝑚 (𝑥 - 𝑥 1)$ .
Distribute and simplify to slope-intercept or standard form as needed.
Verify by substituting the other point into your equation.

Misconceptions

Subtracting coordinates in different orders — using $𝑦 2 - 𝑦 1$ in the numerator but $𝑥 1 - 𝑥 2$ in the denominator flips the sign of the slope.
Reading rise/run from a graph as run/rise — rise is the vertical change (y), run is the horizontal change (x).
Forgetting to distribute the slope through the parentheses in point-slope form before simplifying.

Question

Worked Example

A line passes through the points $(2, 3)$ and $(6, 11)$ . What is the equation of this line in slope-intercept form?

Select an answer to see the explanation