Slope Formula: Rise Over Run and Slope-Intercept Form

The slope formula gives the slope of a line from two points:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Use it when you know two points on the same line and want the line's steepness, or rate of change. In plain language, slope is rise over run: change in $y$ divided by change in $x$.

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

If $m > 0$, the line rises from left to right. If $m < 0$, it falls. If $m = 0$, the line is horizontal.

What the slope formula means

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.

That is why the slope formula and rise over run are the same idea. The formula is just the coordinate version of that ratio.

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 $(x_2, y_2)$.

Start with the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Substitute the coordinates in the same order:

$$m = \frac{9 - 3}{5 - 2} = \frac{6}{3} = 2$$

So the slope is $2$. That means whenever $x$ increases by $1$, $y$ increases by $2$.

You can see the same result as rise over run. From $(2, 3)$ to $(5, 9)$, the rise is $6$ and the run is $3$, so

$$\frac{\text{rise}}{\text{run}} = \frac{6}{3} = 2$$

From slope formula to slope-intercept form

Once you know the slope, you can use slope-intercept form

$$y = mx + b$$

to write the equation of the line, as long as the line is not vertical.

Using the example above, $m = 2$. Substitute one point, such as $(2, 3)$:

$$3 = 2(2) + b$$ $$3 = 4 + b$$ $$b = -1$$

So the line is

$$y = 2x - 1$$

The connection is practical: the slope formula gives you $m$, and slope-intercept form uses that slope to write the full equation.

Common mistakes with the slope formula

One common mistake 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 $x_2 - x_1$.

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

When to use the slope formula

Use the slope formula when you know two points on a line and want its rate of change. This comes up in algebra, coordinate geometry, graphing, and any linear relationship where equal changes in $x$ produce a constant change in $y$.

If the graph is not a straight line, the slope between two points is only the slope of the secant line between those points. It is not one constant slope for the whole graph.

Try a similar problem

Try your own version with the points $(1, -2)$ and $(4, 7)$. Find the slope first, then use one point to write the equation in slope-intercept form. If you want another case right after this, continue with How To Find Slope or Slope Intercept Form.