Slope Formula: Rise Over Run and Slope-Intercept Form

The slope formula turns two points on a line into a single number for its steepness, or rate of change. Use it whenever you know two points on the same line and want to know how fast $y$ changes relative to $x$.

The Formula And Its Symbols

The slope of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is

The numerator $y_2 - y_1$ is the vertical change, also called the rise. The denominator $x_2 - x_1$ is the horizontal change, also called the run. So $m$ is simply rise over run, written in coordinates.

This works only when $x_2 \ne x_1$. If the two points have the same $x$-value, the line is vertical, the denominator is $0$, and the slope is undefined.

The sign of $m$ tells you the direction: if $m > 0$, the line rises from left to right; if $m < 0$, it falls; if $m = 0$, the line is horizontal.

Why It Equals Rise Over Run

The slope formula and "rise over run" are the same idea, not two separate rules. Rise is how far you move vertically between the points, and run is how far you move horizontally. Dividing one by the other measures how steep the path is. The formula just reads those two changes straight off the coordinates, so the coordinate version and the verbal version always agree.

Worked Example: Find Slope From Two Points

Find the slope of the line through $(2, 3)$ and $(5, 9)$. Label the first point as $(x_1, y_1)$ and the second as $(x_2, y_2)$, then substitute in the same order:

So the slope is $2$, meaning whenever $x$ increases by $1$, $y$ increases by $2$. As rise over run, the rise from $(2, 3)$ to $(5, 9)$ is $6$ and the run is $3$, so

Once you know the slope, you can write the full line using slope-intercept form $y = mx + b$. With $m = 2$ and the point $(2, 3)$:

So the line is $y = 2x - 1$.

Practice: Build The Equation Yourself

Use the points $(1, -2)$ and $(4, 7)$. First apply the slope formula, keeping the $y$-differences and $x$-differences in the same order, then substitute the slope and one point into $y = mx + b$ to solve for $b$.

Check your result: the slope should come out to $3$, and the equation should be $y = 3x - 5$. If your slope is right but the equation is off, recheck the value of $b$ you substituted.

Calculation Pitfalls

The most common slip is subtracting the $y$-values in one order and the $x$-values in the opposite order. If you use $y_2 - y_1$, you must also use $x_2 - x_1$.

A second trap is saying a vertical line has slope $0$. A horizontal line has slope $0$; a vertical line has undefined slope because the denominator becomes $0$.

A third is ignoring the sign. A negative slope means the line goes down as $x$ increases.

One more limit: if the graph is not a straight line, the slope between two points is only the slope of the secant line between them, not a single constant slope for the whole graph.

When To Use The Slope Formula

Reach for the slope formula when you know two points on a line and want its rate of change. It appears throughout algebra, coordinate geometry, graphing, and any linear relationship where equal changes in $x$ produce a constant change in $y$.