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 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

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