Writing Equations in Point-Slope Form

Writing an equation in point-slope form comes down to one move: find a point and the slope, then substitute. What changes is where those two pieces come from. Here’s how to handle each starting point.

The Formula You’re Writing Into

Every equation you write lands in the same template, so keep it in front of you:

y − y₁ = m(x − x₁)

From a Point and a Slope

This is the direct case. If you’re given the point (2, 5) and slope 3, just substitute: y − 5 = 3(x − 2). You’re done, no rearranging needed.

From Two Points

When you have two points, add one step at the front: find the slope with m = (y₂ − y₁) / (x₂ − x₁), then substitute either point. For (1, 2) and (4, 8), the slope is 2, so you’d write y − 2 = 2(x − 1). The two-point guide walks through this in detail.

From a Graph

Reading a line off a graph is the same idea. Pick a clear point the line passes through, then count the rise over the run to a second clear point for the slope. If the line goes through (0, 1) and rises 2 for every 1 across, that’s y − 1 = 2(x − 0). See how to graph point-slope form for the reverse skill.

From a Slope and Y-Intercept

The y-intercept is a point too, sitting at (0, b). Use it as your point. With slope 4 and intercept −3, you get y − (−3) = 4(x − 0), which simplifies to y = 4x − 3.

Common Slip-Ups

• Sign errors. A point of (3, −2) gives y + 2, because subtracting a negative flips to a plus.
• Skipping the slope. With two points, you can’t write the equation until you’ve found m.
• Substituting into x and y. Only x₁ and y₁ get numbers.

Check any equation you write with the calculator, or see it applied in worked examples.