Straight Lines

Quick Reference

Concept	Formula	Notes
Slope	$m=\frac{y_2-y_1}{x_2-x_1}=\frac{\text{rise}}{\text{run}}$	Ratio of vertical change to horizontal change
Point-Slope Form	$y-y_1=m(x-x_1)$	Given slope m and point $(x_1,y_1)$
Point-Point Form	$y=y_1+\left(\frac{y_2-y_1}{x_2-x_1}\right)(x-x_1)$	Given two points $(x_1,y_1)$ and $(x_2,y_2)$
Parallel Lines	$m_1=m_2$	Equal slopes
Perpendicular Lines	$m_1{m}_2=-1$	Slopes are negative reciprocals

Slope of a Line

Consider a straight line L and two distinct points $P(x_1,y_1)$ and $Q(x_2,y_2)$ on it (see the following figure). The slope of the line, often denoted by m, is defined to be the ratio

• If $\alpha$ ( or ) is the angle that a non-horizontal line makes with the positive direction of the x-axis (see the above figure), then $m=\tan \alpha$.
• The sign of the slope is related to the direction of the line as follows:
  • If , $\alpha$ is an acute angle, and the line rises to the right.
  • If , $\alpha$ is an obtuse angle, and the line falls to the right.
  • If $m=0$, the line is horizontal.

The equation of the line passing through a given point $P(x_1,y_1)$ with slope m is

This equation is called the point-slope form of the equation of the line.

If we are given two points $(x_1,y_1)$ and $(x_2,y_2)$ on the line, then to find the equation of the line, we first have to calculate m, which is $m=(y_2-y_1)/(x_2-x_1)$. Substitution of this expression for m in the point-slope form of the equation gives the point-point form of the equation of the line:

$$y=y_1+\left(\frac{y_2-y_1}{x_2-x_1}\right)(x-x_1)$$

• The equation of a vertical line through $(x_1,y_1)$ is $x=x_1$.

Parallel and Perpendicular Lines

Two lines with slopes $m_1$ and $m_2$ are

• parallel $⟺m_1=m_2$.
• perpendicular $⟺m_1{m}_2=-1$.

It is clear that horizontal lines (slope 0) are perpendicular to vertical lines (no slope).