Overview

Lines are the simplest coordinate objects. Many geometry problems reduce to finding a line and intersecting it with another curve.

Key Ideas

• Point-slope: $y-y_1 = m(x-x_1)$.
• Slope-intercept: $y = mx + b$.
• Perpendicular slopes satisfy $m_1 m_2 = -1$.
• Horizontal lines: $y=c$. Vertical lines: $x=c$.

Core Skills

Choose the Best Form

Use point-slope when you know a point and slope, slope-intercept for quick graphing, and standard form when working with intercepts.

Find Intersections

Set the equations equal or solve the system to find intersection points. This is the standard way to link lines to geometry.

Handle Vertical Lines

Vertical lines have undefined slope and must be written as $x=c$.

Worked Example

Line $\ell_1$ has slope $2$. Find the slope of a perpendicular line.

$m_2 = -1/2$.

More Examples

Example 1: Point-Slope

Find the equation of the line through $(3,1)$ with slope $-4$.

$y-1 = -4(x-3)$.

Example 2: Two Points

Find the line through $(1,2)$ and $(5,6)$.

Slope $m=(6-2)/(5-1)=1$, so $y-2 = 1(x-1)$ or $y=x+1$.

Example 3: Vertical Line

Find the equation of the line through $(2,-3)$ with undefined slope.

$x=2$.

Strategy Checklist

• Compute the slope first unless the line is vertical.
• Pick the form that matches the given data.
• Check intersection points by substitution.

Common Pitfalls

• Mixing up negative reciprocal with negative slope.
• Using vertical lines in slope form (slope undefined).
• Dropping a sign when distributing in point-slope form.

Practice Problems