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Amy Ferguson Moncure

H Algebra 2/Trig · Unit 1

Write Equations of Lines

Write y = mx + c from two points, from a parallel or perpendicular condition, or find a missing coordinate from a given slope.

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Key Ideas Easy Medium Hard

m = y₂ − y₁x₂ − x₁

y − y₁ = m(x − x₁)

What to Remember

Two points

First find the slope: m = y₂ − y₁x₂ − x₁. Then use either point in point-slope form.

Parallel

Keep the same slope. Use the given point to find c.

Perpendicular

Use the negative reciprocal slope. Then use the given point.

Missing coordinate

Plug the known slope and the two points into the slope formula, then solve for the missing value.

Level 1

Easy

Two points. Write y = mx + c.

Example
Write the equation of the line through (1, 2) and (5, 10).

The formula is

m = y₂ − y₁x₂ − x₁

Substitute.

m = 10 − 25 − 1

m = 84

m = 2

The formula is

y − y₁ = m(x − x₁)

Substitute.

y − 2 = 2(x − 1)

y = 2x

Answer: y = 2x

1
Write the equation of the line through (0, 4) and (3, −2).
Show Answer

The formula is

m = y₂ − y₁x₂ − x₁

Substitute.

m = −2 − 43 − 0

m = −63

m = −2

The formula is

y − y₁ = m(x − x₁)

Substitute.

y − 4 = −2(x − 0)

y = −2x + 4

Answer: y = −2x + 4

2
Write the equation of the line through (−1, 6) and (3, 6).
Show Answer

The formula is

m = y₂ − y₁x₂ − x₁

Substitute.

m = 6 − 63 − (−1)

m = 04

m = 0

The formula is

y − y₁ = m(x − x₁)

Substitute.

y − 6 = 0(x − (−1))

y = 6

Answer: y = 6

3
Write the equation of the line through (2, −3) and (6, 5).
Show Answer

The formula is

m = y₂ − y₁x₂ − x₁

Substitute.

m = 5 − (−3)6 − 2

m = 84

m = 2

The formula is

y − y₁ = m(x − x₁)

Substitute.

y − (−3) = 2(x − 2)

y = 2x − 7

Answer: y = 2x − 7

4
Write the equation of the line through (−4, 1) and (0, 9).
Show Answer

The formula is

m = y₂ − y₁x₂ − x₁

Substitute.

m = 9 − 10 − (−4)

m = 84

m = 2

The formula is

y − y₁ = m(x − x₁)

Substitute.

y − 1 = 2(x − (−4))

y = 2x + 9

Answer: y = 2x + 9

5
Write the equation of the line through (0, −5) with slope 4.
Show Answer

The given point is the y-intercept.

The formula is

y = mx + c

y = 4x − 5

Answer: y = 4x − 5

Level 2

Medium

A line parallel to a given line, through a given point.

Example
Line AB has slope −2. Write the equation of the line parallel to AB through (4, 1).

Parallel lines have the same slope.

m = −2

The formula is

y − y₁ = m(x − x₁)

Substitute.

y − 1 = −2(x − 4)

y = −2x + 9

Answer: y = −2x + 9

1
Write the equation of the line through (−2, 5) parallel to y = 3x + 1.
Show Answer

Parallel lines have the same slope.

m = 3

The formula is

y − y₁ = m(x − x₁)

Substitute.

y − 5 = 3(x − (−2))

y = 3x + 11

Answer: y = 3x + 11

2
Write the equation of the line through (6, −1) parallel to y = 12x − 4.
Show Answer

Parallel lines have the same slope.

m = 12

The formula is

y − y₁ = m(x − x₁)

Substitute.

y − (−1) = 12(x − 6)

y = 12x − 4

Answer: y = 12x − 4

3
Write the equation of the line through (1, 0) parallel to y = −4x + 7.
Show Answer

Parallel lines have the same slope.

m = −4

The formula is

y − y₁ = m(x − x₁)

Substitute.

y − 0 = −4(x − 1)

y = −4x + 4

Answer: y = −4x + 4

4
Write the equation of the line through (2, −7) parallel to y = 5x − 3.
Show Answer

Parallel lines have the same slope.

m = 5

The formula is

y − y₁ = m(x − x₁)

Substitute.

y − (−7) = 5(x − 2)

y = 5x − 17

Answer: y = 5x − 17

5
Given A(2, 5) and B(2, 11). Write the equation of line AB.
Show Answer

Both points have x = 2.

The line is vertical.

x = 2

Answer: x = 2

Level 3

Hard

Perpendicular through a point, or a missing coordinate from a given slope.

Example
A line has slope 2. Write the equation of the line perpendicular to it through (−6, 4).

Perpendicular slopes are negative reciprocals.

The formula is

m⊥ = −1m

Substitute.

−12

The formula is

y − y₁ = m(x − x₁)

Substitute.

y − 4 = −12(x − (−6))

y = −12x + 1

Answer: y = −12x + 1

1
Write the equation of the line through (9, 2) perpendicular to y = −3x + 1.
Show Answer

Perpendicular slopes are negative reciprocals.

The formula is

m⊥ = −1m

Substitute.

13

The formula is

y − y₁ = m(x − x₁)

Substitute.

y − 2 = 13(x − 9)

y = 13x − 1

Answer: y = 13x − 1

2
Write the equation of the line through (−8, 1) perpendicular to y = 14x − 2.
Show Answer

Perpendicular slopes are negative reciprocals.

The formula is

m⊥ = −1m

Substitute.

−4

The formula is

y − y₁ = m(x − x₁)

Substitute.

y − 1 = −4(x − (−8))

y = −4x − 31

Answer: y = −4x − 31

3
The slope between A(1, y) and B(5, 11) is 2. Find y.
Show Answer

The formula is

m = y₂ − y₁x₂ − x₁

Substitute.

2 = 11 − y5 − 1

2 = 11 − y4

2(4) = 11 − y

8 = 11 − y

y = 3

Answer: 3

4
The slope between A(−2, y) and B(4, 10) is 3. Find y.
Show Answer

The formula is

m = y₂ − y₁x₂ − x₁

Substitute.

3 = 10 − y4 − (−2)

3 = 10 − y6

3(6) = 10 − y

18 = 10 − y

y = −8

Answer: −8

5
The slope between A(4, y) and B(−2, 8) is −1. Find y.
Show Answer

The formula is

m = y₂ − y₁x₂ − x₁

Substitute.

-1 = 8 − y−2 − 4

-1 = 8 − y−6

−1(−6) = 8 − y

6 = 8 − y

y = 2

Answer: 2

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