The Slope Formula

The formula, piece by piece

Call your two points $(x_1, y_1)$ and $(x_2, y_2)$. The slope is $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ — the change in $y$ on top, the change in $x$ on the bottom.

The top, $y_2 - y_1$, tells you how far the line moved vertically between the two points. The bottom, $x_2 - x_1$, tells you how far it moved horizontally. Dividing gives the vertical change per one unit of horizontal change — that's the intuition people call rise over run, but the formula itself is always $\dfrac{y_2 - y_1}{x_2 - x_1}$.

In the graph below, the line passes through $(1, 1)$ and $(3, 5)$. Between those points, $y$ changed by $5 - 1 = 4$ while $x$ changed by $3 - 1 = 2$, so $m = \dfrac{4}{2} = 2$.

Keep the order consistent

It does not matter which point you call $(x_1, y_1)$ — but once you choose, you must subtract in the same order on top and bottom. If you write $y_2 - y_1$ on top, the bottom must be $x_2 - x_1$, not $x_1 - x_2$. Mixing the order flips the sign of your answer, and a wrong sign means a line that tilts the wrong way.

Negative coordinates are where most slope mistakes happen. Subtracting a negative becomes addition: with points $(-3, 4)$ and $(2, -6)$, the bottom is $2 - (-3) = 5$, not $-1$. Write out the substitution with parentheses before simplifying anything.

Zero slope and undefined slope

If the two points have the same $y$-value, the top of the formula is $0$, so the slope is $0$ — a horizontal line. Zero slope is a perfectly good slope: the line is flat, not missing.

If the two points have the same $x$-value, the bottom of the formula is $0$, and dividing by zero is not allowed — the slope is undefined. That's a vertical line. Keep these two straight: horizontal means slope $0$, vertical means no slope at all.

Worked examples

Try one yourself

Common questions