A straight line has a constant rate of change: whenever $x$ changes by the same amount, $y$ changes by a proportional amount. That rate is the slope.

For two points $(x_1,y_1)$ and $(x_2,y_2)$ on a nonvertical line,

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

Interpreting slope

A positive slope rises from left to right, a negative slope falls, and zero slope gives a horizontal line. A vertical line has $\Delta x=0$, so its slope is undefined.

For example, the line through $(1,2)$ and $(4,8)$ has

$$m=\frac{8-2}{4-1}=2.$$

This means that $y$ increases by $2$ for each increase of $1$ in $x$.

Equations of lines

A line with slope $m$ and vertical intercept $b$ can be written

$$y=mx+b.$$

If a point $(x_1,y_1)$ and the slope are known, the point-slope form is

$$y-y_1=m(x-x_1).$$

Both equations describe the same geometric object in different ways.

Parallel and perpendicular lines

Distinct nonvertical parallel lines have the same slope. Perpendicular nonvertical lines have slopes whose product is $-1$ when both slopes are defined, so their slopes are negative reciprocals.