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Daily Math Minute

Linear Functions

Point-Slope Form

Writing the equation of a line given a point and the slope.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

  • Slope-Intercept Form

Writing an Equation Without Knowing the Y-Intercept

A line has a slope of 4 and passes through (3, 5) — but that point isn't the y-intercept, and finding it would take an extra step. Before reading on, think back to the slope formula itself: could it be rearranged to build an equation directly from just one point and the slope, without ever solving for b first?

Definition — Point-Slope Form

A linear equation written as y − y₁ = m(x − x₁), built from a known slope m and a known point (x₁, y₁) on the line.

This form comes directly from the slope formula. Starting from m = (y − y₁)/(x − x₁) — the slope between the known point and any general point (x, y) on the line — multiplying both sides by (x − x₁) clears the fraction and leaves exactly y − y₁ = m(x − x₁). It's not a new idea, just the slope formula rearranged to isolate y − y₁ instead of m.

Worked Example — Writing an Equation from a Point and a Slope

Write the equation of the line through (3, 5) with slope 4. Substitute directly into point-slope form: y − 5 = 4(x − 3). This can be converted to slope-intercept form by distributing and isolating y: y − 5 = 4x − 12, so y = 4x − 7.

Worked Example — Writing an Equation from Two Points

Write the equation of the line through (2, 1) and (6, 13). First find the slope: m = (13 − 1)/(6 − 2) = 12/4 = 3. Use either point in point-slope form, say (2, 1): y − 1 = 3(x − 2). Simplify: y = 3x − 5.
y−y1=m(x−x1)y - y_1 = m(x - x_1)

Graph Visualizer

Domain & range
2
Evaluate a point
  • x^2 = 0

Tip

Either point on a line works equally well in point-slope form — using the other available point in the two-points example above still produces the exact same final equation, just by a different path.

Common Mistakes

  • Substituting the point's coordinates in the wrong position, such as writing y − x₁ = m(x − y₁), swapping which coordinate goes where.

    The point-slope form keeps y with y₁ and x with x₁ — match each coordinate to its own variable before substituting.

  • Leaving the equation in point-slope form when a problem specifically asks for slope-intercept form, without distributing and isolating y.

    Point-slope form is a valid equation on its own, but converting to y = mx + b requires distributing the m and then isolating y — check which form the problem actually wants.

Key Takeaways

  • Point-slope form, y − y₁ = m(x − x₁), builds a line's equation from any one point and the slope.
  • This form comes directly from rearranging the slope formula itself.
  • Given two points, find the slope first, then use either point in point-slope form.

Summary

Point-slope form removes the need to know a line's y-intercept before writing its equation. The next lesson uses slope comparisons to write equations for lines with a very specific relationship to a given line — parallel or perpendicular to it.

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