# Equations of Parallel and Perpendicular Lines

## The Basics

Review: the equation of a line is written as **y = mx + b**

Where **m** is the slope

And **b** is the y-intercept

There is also point-slope form: y - y1 = m(x - x1)

But be sure to convert to y = mx + b for simplicity and consistency

## Parallel Lines

Parallel lines have the same slope (m) and different y-intercepts (b)

Therefore, lines with the same slope and y-intercepts cannot be "parallel" they are simply the same line

Take a look at the example on the left. Let's say they are parallel. Find the equation.

Step 1: write the equation of the blue line: **y = 2x+1**

Step 2: we know the yellow line's equation will be **y = 2x + b** if they are parallel

Step 3: plug in y and x to solve for b:** 4 = 2(5) + b** and therefore **b = -6**

Step 4: yellow line's equation: **y = 2x - 6 **

## Perpendicular Lines

This is a bit trickier! Perpendicular lines have **opposite and reciprocal slopes**. What does that mean???

Opposite: if one slope is positive, the other is negative, and vice versa. All you have to do is negate one or the other.

Reciprocal: if one slope is m, the other is 1/m. For example, if the slope of one is -2, the other would be 1/2 (opposite AND reciprocal!)

In the case of perpendicular lines, the y-intercept does not matter. They can be the same.

Let's take a look at an example on the right.

Step 1: write down the blue line's equation: **y = -4x + 10**

Step 2: find the slope of the yellow line: remember, it's the opposite reciprocal so the answer is 1/4

Step 3: write yellow line's equation and plug in the points to find the y-intercept: **2 = 1/4(7) + b**, therefore **b = 1/4**

Step 4: the final equation: **y = 1/4x + 1/4**