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Amy Ferguson Moncure

Algebra 1 · Unit 6

Parallel and Perpendicular Lines

Write the equation of a line through a given point that is parallel or perpendicular to a given line.

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Key Ideas Easy Medium Hard Super Challenge
What to Remember

Parallel

Parallel lines have the same slope and different intercepts.

Perpendicular

Perpendicular slopes multiply to −1. The slopes are negative reciprocals.

Through a point

Keep the slope you need. Plug the given point into y = mx + b to find b, or use point-slope.

Vertical and horizontal

A horizontal line has slope 0. A vertical line has undefined slope. They are perpendicular to each other.

Level 1

Easy

Same slope or negative reciprocal. Integer slopes.

Example
Write the line parallel to y = 2x + 1 through (0, 5).

Parallel slope is 2.

Through (0, 5), b = 5.

y = 2x + 5

Answer: y = 2x + 5

1
Write the line parallel to y = −3x + 4 through (0, −1).
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y = −3x − 1

Answer: y = −3x − 1

2
Write the line perpendicular to y = 12x through (0, 3).
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Perpendicular slope is −2.

y = −2x + 3

Answer: y = −2x + 3

3
Write the horizontal line through (4, −2).
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y = −2

Answer: y = −2

4
Write the vertical line through (4, −2).
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x = 4

Answer: x = 4

5
Are y = 3x − 1 and y = 3x + 8 parallel, perpendicular, or neither?
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Same slope 3, different intercepts. Parallel.

Answer: parallel

Level 2

Medium

The given line is not through the origin. Solve for b.

Example
Write the line parallel to y = −1x + 6 through (2, 3).

m = −1

3 = −1(2) + b

3 = −2 + b

b = 5

y = −1x + 5

Answer: y = −x + 5

1
Write the line perpendicular to y = 4x − 1 through (4, 2).
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m = −14

2 = −14(4) + b

2 = −1 + b

b = 3

y = −14x + 3

Answer: y = −14x + 3

2
Write the line parallel to 2x + y = 7 through (3, −4).
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y = −2x + 7, so m = −2

−4 = −2(3) + b

b = 2

y = −2x + 2

Answer: y = −2x + 2

3
Write the line perpendicular to y = −3x + 2 through (0, 0).
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m = 13

y = 13x

Answer: y = 13x

4
A line through (1, 5) is parallel to the x-axis. Write it.
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y = 5

Answer: y = 5

5
Write the line perpendicular to x = 2 through (5, −1).
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x = 2 is vertical, so perpendicular is horizontal.

y = −1

Answer: y = −1

Level 3

Hard

Standard form, or a slope that is already a fraction.

Example
Write the line perpendicular to 3x − 6y = 12 through (2, −1).

−6y = −3x + 12

y = 12x − 2

Perpendicular slope −2

−1 = −2(2) + b

b = 3

y = −2x + 3

Answer: y = −2x + 3

1
Write the line parallel to y = 23x − 5 through (−3, 4).
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m = 23

4 = 23(−3) + b

4 = −2 + b

b = 6

y = 23x + 6

Answer: y = 23x + 6

2
Write the line perpendicular to x + 2y = 8 through (4, 0).
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y = −12x + 4

Perpendicular slope 2

0 = 2(4) + b

b = −8

y = 2x − 8

Answer: y = 2x − 8

3
Find k so y = kx + 1 is perpendicular to y = 5x − 3.
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k × 5 = −1

k = −15

Answer: −15

4
Write the line through (3, 3) that is parallel to the y-axis.
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x = 3

Answer: x = 3

5
Two lines: y = mx + 2 and y = (m + 1)x − 4. For which m are they perpendicular?
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m(m + 1) = −1

m² + m + 1 = 0

Discriminant 1 − 4 = −3 < 0, so no real m.

Answer: no real m

Super Challenge

Super Challenge

1
Write the line through (1, 2) that is perpendicular to the line through (1, 2) and (4, 8).
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Given slope 8 − 24 − 1 = 2

Perpendicular slope −12

2 = −12(1) + b

b = 52

y = −12x + 52

Answer: y = −12x + 52

2
A student says any two lines with slopes 2 and 12 are perpendicular. What is missing?
Show Answer

2 × 12 = 1, not −1.

They are not negative reciprocals. Perpendicular would be 2 and −12.

Answer: product must be −1

3
Write two different lines through (0, 4): one parallel to y = x, one perpendicular to y = x.
Show Answer

Parallel: y = x + 4

Perpendicular: y = −1x + 4

Answer: y = x + 4 and y = −x + 4

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