Slope Calculator

The slope formula

The slope $m$ of the line through $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

This is "rise over run": $\Delta y$ is how far the line rises (or falls) vertically, and $\Delta x$ is how far it runs horizontally, between the two points.

Two special cases

• Horizontal line ($\Delta y = 0$): the slope is exactly $0$ — $y$ never changes.
• Vertical line ($\Delta x = 0$): the slope is undefined — division by zero has no value. A vertical line's equation is simply $x = x_1$.

These are opposite situations, not the same: a $0$ slope is a perfectly real, flat line; an undefined slope means there is no single rate of rise per unit of run, because the run itself is zero.

Three equations for the same line

Once the slope $m$ and a point $(x_1,y_1)$ are known, the line can be written three ways:

Point-slope form: $y - y_1 = m(x - x_1)$

Slope-intercept form: solve point-slope for $y$ to get $y = mx + b$, where $b = y_1 - mx_1$ is the $y$-intercept.

Standard form: clear the fractions in slope-intercept form and move every term to one side: $Ax + By = C$, with $A$, $B$, $C$ integers, $A \geq 0$, and no common factor other than $1$.

Worked example

For $(2, 3)$ and $(7, -9)$: $\Delta y = -9-3 = -12$ and $\Delta x = 7-2 = 5$, so $m = -\frac{12}{5} = -2.4$.

Point-slope through $(2,3)$: $y - 3 = -\frac{12}{5}(x-2)$.

Solving for $y$: $y = -\frac{12}{5}x + \frac{39}{5}$ (so $b = \frac{39}{5} = 7.8$).

Clearing denominators: $12x + 5y = 39$.

The line crosses the $y$-axis at $\left(0, \frac{39}{5}\right)$ and the $x$-axis at $\left(\frac{13}{4}, 0\right)$ — found by setting $x=0$ (giving $y=b$) and $y=0$ (giving $x=-\frac{b}{m}$) in slope-intercept form.

Parallel and perpendicular slopes

A line parallel to this one has the identical slope $m$. A line perpendicular to it has the negative reciprocal slope $-\frac{1}{m}$ — for $m=-\frac{12}{5}$, that's $\frac{5}{12}$. A horizontal line ($m=0$) is perpendicular to a vertical one (undefined slope), and vice versa.

Related calculators

The same two points also determine the Distance Between Two Points between them (the straight-line length of the segment) — this calculator instead finds how steep that segment is, and every equation of the line through it. Slope and the Quadratic Formula both describe curves from their coefficients; a line is simply the degree-1 case ($m$ and $b$ instead of $a$, $b$, and $c$).