Skill: Write the equation of a perpendicular line

To write the equation of a line perpendicular to a given line:

1. Find the slope of the given line.
2. Find its negative reciprocal:
   • Change the sign.
   • Flip the fraction.
3. Use the new slope and the given point in point-slope form:

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

Example

Find the equation of the line perpendicular to

$$y=\frac{2}{3}x-5$$

that passes through $(6,1)$.

Step 1: Identify the given slope.

The equation is in slope-intercept form, $y=mx+b$, so the slope is

$$m=\frac{2}{3}.$$

Step 2: Find the perpendicular slope.

The negative reciprocal of $\frac{2}{3}$ is

$$m_{\perp}=-\frac{3}{2}.$$

Step 3: Use point-slope form.

The line must pass through $(6,1)$, so substitute $x_1=6$, $y_1=1$, and $m=-\frac{3}{2}$:

$$y-1=-\frac{3}{2}(x-6).$$

This is the equation of the perpendicular line in point-slope form.

To write it in slope-intercept form, simplify:

$$y-1=-\frac{3}{2}x+9$$ $$y=-\frac{3}{2}x+10.$$

Therefore, the perpendicular line is

$$\boxed{y=-\frac{3}{2}x+10}.$$

Remember the special cases:

• A horizontal line has slope $0$, so a perpendicular line is vertical: $x=c$.
• A vertical line has an undefined slope, so a perpendicular line is horizontal: $y=c$.