Algebra I

Parallel and Perpendicular Lines

Lesson

Two lines on the plane can have a special relationship — and you can read it off their slopes.

Parallel lines

m_1 = m_2

Same slope. They never meet. (If they have the same y-intercept too, they’re the same line, not parallel.)

Perpendicular lines

m_1 \cdot m_2 = -1

Their slopes are negative reciprocals. Flip the fraction and switch the sign. Example: if $m_1 = \tfrac{2}{3}$ , then $m_2 = -\tfrac{3}{2}$ .

Worked example 1 — parallel

A line is parallel to $y = 4x + 7$ . Its slope is $4$ .

Worked example 2 — perpendicular

A line is perpendicular to $y = -\tfrac{1}{2}x + 3$ . Negative reciprocal of $-\tfrac{1}{2}$ is $2$ . So the slope is $2$ .

Worked example 3 — through a point

Find the line parallel to $y = 3x + 1$ that passes through $(2, 5)$ .

Slope is $3$ . Use point-slope:

y - 5 = 3(x - 2) \implies y = 3x - 1

Quick check

If you multiply the two slopes and get $-1$ , the lines are perpendicular. If they match, the lines are parallel. Otherwise neither.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

\text{Slope of any line parallel to } y = 2x + 5

Problem 2

\text{Slope of any line parallel to } y = -3x + 1

Problem 3

\text{Slope of any line perpendicular to } y = 2x + 5

Problem 4

\text{Slope of any line perpendicular to } y = -4x

Practice

Standard problems matching the lesson.

Problem 5

\text{Slope of a line parallel to } y = 5x - 2

Problem 6

\text{Slope of a line perpendicular to } y = 5x - 2

Problem 7

\text{Slope of a line parallel to } y = \tfrac{2}{3}x + 1

Problem 8

\text{Slope of a line perpendicular to } y = \tfrac{2}{3}x + 1

Problem 9

\text{Slope of a line parallel to } 3x + 4y = 12

Problem 10

\text{Slope of a line perpendicular to } 3x + 4y = 12

Problem 11

\text{Slope of a line parallel to } y = -7x + 4

Problem 12

\text{Slope of a line perpendicular to } y = \tfrac{1}{2}x

Problem 13

Line parallel to y = 4x + 1 through (0, 7). y-intercept?

Problem 14

Line perpendicular to y = 2x + 3 through (0, -5). y-intercept?

Problem 15

\text{Slope of a line perpendicular to } y = -x + 5

Problem 16

Line parallel to 2x - y = 7 through (0, 10). y-intercept?

Problem 17

Two parallel streets. One has slope 5/8. Other slope?

Problem 18

Flagpole perpendicular to a ramp of slope 2/3. Slope of flagpole?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

\text{Slope of a line perpendicular to } 2x + 3y = 6

Problem 20

\text{Slope of a line parallel to } -5x + 2y = 8

Problem 21

Line through (2, 5) parallel to y = 3x. y-intercept?

Problem 22

Line through (4, 1) perpendicular to y = 2x. y-intercept?

Problem 23

\text{For } y = mx \text{ to be perpendicular to } y = 2x, \ m = ?

Problem 24

\text{For } y = mx + 1 \text{ to be parallel to } y = 4x - 3, \ m = ?

Problem 25

Slope perpendicular to line through (0,0) and (4,8)?

Ask the tutor

Stuck on a concept? Want another example? Ask anything about this topic.

Type your own question below, or tap one of the suggestions. The tutor can re-explain the lesson, work through a specific problem with you, generate fresh practice tuned to where you are, or check your reasoning.

Quiz

Test yourself on this topic →

10 questions, no hints. About 5 minutes.