Another viewpoint

Point-slope form: write a line from one point and a slope

Start with a known point before rearranging into y = mx + b.

Point-slope form is useful when you know a slope m and one point (x₁, y₁). It describes how every other point (x, y) on the line differs from that known point.

It comes directly from the slope formula

The relationship m = (y − y₁)/(x − x₁) can be rearranged to y − y₁ = m(x − x₁) for points where x differs from x₁. The resulting linear equation also includes the original point, because substituting x = x₁ and y = y₁ gives 0 = 0.

Example: slope 2 through (3, 4)

Substitute the known point and slope: y − 4 = 2(x − 3). This is already a complete equation of the line. If you need slope-intercept form, expand and rearrange:

1. Distribute 2: y − 4 = 2x − 6.
2. Add 4 to both sides: y = 2x − 2.
3. Check the known point: 4 = 2 × 3 − 2.

Be careful with negative coordinates

For slope -1/2 through (-2, 5), write y − 5 = (-1/2)(x − (-2)). The inner expression becomes x + 2, so y − 5 = (-1/2)(x + 2). Expanding gives y = (-1/2)x + 4.

The sign changes because subtracting a negative number means adding its positive counterpart. Keeping the original substitution visible until the next line makes this less error-prone.

Which form should you use?

Point-slope form is convenient for building an equation from a point and a slope. Slope-intercept form is convenient for reading the intercept and evaluating y at a chosen x. Neither is more correct; they can describe exactly the same nonvertical line.

A vertical line needs x = constant because its slope is undefined. A horizontal line works with m = 0 and simplifies to y = y₁. Review the special cases if these look similar.

When starting with two points, first use the slope formula, then substitute one point. The calculator shows the same slope and the equivalent slope-intercept equation.