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Amy Ferguson Moncure

Algebra 1 · Unit 2

Parallel and Perpendicular Lines

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

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Key Ideas Easy Medium Hard Super Challenge
What to Remember

Parallel lines

Parallel lines have the same slope and never meet. Vertical lines are parallel to each other. Horizontal lines are parallel to each other.

Perpendicular lines

Perpendicular lines meet at a right angle. Their slopes are negative reciprocals: if one slope is ab, the other is −1ba. A vertical line is perpendicular to a horizontal line.

Through a given point

Keep the slope you need (same, or negative reciprocal). Then use the given point in point-slope form.

Check

Compare slopes, then substitute the given point to make sure it lies on your new line.

Level 1

Easy

Read slopes, then say parallel, perpendicular, or neither.

Example
Are y = 3x + 1 and y = 3x − 8 parallel, perpendicular, or neither?

Both slopes are 3.

Same slope, different intercepts, so the lines are parallel.

Answer: parallel

1
Are y = 2x + 5 and y = −12x + 1 parallel, perpendicular, or neither?
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Slopes 2 and −12

2 ×−12 = −1, so they are perpendicular.

Answer: perpendicular

2
Write the equation of the line through (0, 4) parallel to y = 5x − 2.
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Same slope 5

Y-intercept 4

y = 5x + 4

Answer: y = 5x + 4

3
Write the equation of the line through (0, −3) perpendicular to y = 13x.
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Perpendicular slope is −3.

y = −3x − 3

Answer: y = −3x − 3

4
A horizontal line through (2, 7) is parallel to which axis? Write its equation.
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Parallel to the x-axis.

y = 7

Answer: y = 7

5
A vertical line through (6, −1) is perpendicular to y = 3. Write its equation.
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y = 3 is horizontal.

Perpendicular is vertical: x = 6.

Answer: x = 6

Level 2

Medium

Through a point that is not the y-intercept.

Example
Write the equation of the line through (2, 5) parallel to y = −4x + 1.

Parallel slope is −4.

y − 5 = −4(x − 2)

y − 5 = −4x + 8

y = −4x + 13

Answer: y = −4x + 13

1
Write the equation of the line through (3, −1) perpendicular to y = 23x + 5.
Show Answer

Perpendicular slope is −32.

y + 1 = −32(x − 3)

y + 1 = −32x + 92

y = −32x + 72

Answer: y = −32x + 72

2
Write the equation of the line through (−2, 1) parallel to 3x + y = 7.
Show Answer

y = −3x + 7, so m = −3.

y − 1 = −3(x + 2)

y − 1 = −3x − 6

y = −3x − 5

Answer: y = −3x − 5

3
Are 2x + 4y = 8 and y = −12x + 9 parallel, perpendicular, or neither?
Show Answer

2x + 4y = 8 → y = −12x + 2

Both slopes are −12.

Different intercepts, so parallel.

Answer: parallel

4
Write the line through (4, 0) perpendicular to x = −2.
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x = −2 is vertical.

Perpendicular is horizontal.

Through (4, 0): y = 0

Answer: y = 0

5
A sidewalk is modeled by y = 14x + 2. A second path through (8, 1) is parallel to the sidewalk. Write its equation.
Show Answer

m = 14

y − 1 = 14(x − 8)

y − 1 = 14x − 2

y = 14x − 1

Answer: y = 14x − 1

Level 3

Hard

Standard form, and a check that the point lies on the new line.

Example
Write the equation of the line through (1, −3) perpendicular to 4x − 2y = 10, in slope-intercept form.

4x − 2y = 10

−2y = −4x + 10

y = 2x − 5

Original slope 2, so perpendicular slope is −12.

y + 3 = −12(x − 1)

y + 3 = −12x + 12

y = −12x − 52

Check (1, −3): −12(1) − 52 = −3.

Answer: y = −12x − 52

1
Write, in standard form with integer coefficients, the line through (6, 2) parallel to x − 3y = 9.
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x − 3y = 9 → −3y = −1x + 9 → y = 13x − 3

Parallel slope 13

y − 2 = 13(x − 6)

3y − 6 = x − 6

−x + 3y = 0

x − 3y = 0

Answer: x − 3y = 0

2
Are y = 5 and x = 5 parallel, perpendicular, or neither?
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y = 5 is horizontal. x = 5 is vertical.

A horizontal line is perpendicular to a vertical line.

Answer: perpendicular

3
Find k so that y = kx + 4 is perpendicular to y = 6x − 1.
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Need k × 6 = −1.

k = −16

Answer: k = −16

4
A roof line is y = 25x + 1. A brace through (10, 3) is perpendicular to the roof. Write the brace’s equation.
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Perpendicular slope −52

y − 3 = −52(x − 10)

y − 3 = −52x + 25

y = −52x + 28

Check: −52(10) + 28 = −25 + 28 = 3

Answer: y = −52x + 28

5
Two lines have slopes 43 and 34. Are they perpendicular? What would the second slope need to be?
Show Answer

43×34 = 1, not −1.

They are not perpendicular (and not parallel).

The perpendicular slope would be −34.

Answer: neither; need −34

Super Challenge

Super Challenge

1
Show that if a line has slope ab with a and b not 0, then a perpendicular line has slope −1ba by multiplying the two slopes.
Show Answer

ab×−1ba = −1

The product of the slopes is −1, so the lines are perpendicular (neither is vertical).

Answer: product −1

2
Find the equation of the line that is perpendicular to 3x − 4y = 12 and has the same y-intercept as 3x − 4y = 12.
Show Answer

3x − 4y = 12 → y = 34x − 3

Y-intercept (0, −3). Original slope 34.

Perpendicular slope −43

y = −43x − 3

Answer: y = −43x − 3

3
A family of lines is y = 2x + b. Describe the line through (1, 4) that is perpendicular to every line in the family. Why is there only one such line?
Show Answer

Every line in the family has slope 2, so they are all parallel to each other.

Perpendicular slope is −12.

Through (1, 4): y − 4 = −12(x − 1)

y = −12x + 92

All the family members share one perpendicular direction, so one slope through one point names one line. That line is perpendicular to every y = 2x + b.

Answer: y = −12x + 92

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