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Amy Ferguson Moncure

Algebra 1 · Unit 2

Write Ax + By = C

Write a line in standard form from intercepts, a slope and a point, or two points.

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Key Ideas Easy Medium Hard
What to Remember

Get to y = mx + b

Find the slope and the y-intercept first. Then move the x term to write Ax + By = C.

Clear fractions

Multiply by a common denominator so A, B, and C are integers.

A positive

If A is negative, multiply both sides by −1. A vertical line is x = a.

Easy

Two intercepts

You are given the x-intercept and the y-intercept.

Example
Write Ax + By = C for the line with x-intercept 4 and y-intercept −2.

The intercepts are (4, 0) and (0, −2).

m = y₂ − y₁x₂ − x₁

m = 24

m = 12

The y-intercept is (0, −2).

y = 12x − 2

Subtract 12x from both sides.

−12x + y = −2

Multiply by 2.

−x + 2y = −4

A is negative. Multiply by −1.

x − 2y = 4

Answer: x − 2y = 4

1
Write Ax + By = C for the line with x-intercept 5 and y-intercept 5.
Show Answer

The intercepts are (5, 0) and (0, 5).

m = y₂ − y₁x₂ − x₁

m = −55

m = −1

The y-intercept is (0, 5).

y = −x + 5

Add x to both sides.

x + y = 5

Answer: x + y = 5

2
Write Ax + By = C for the line with x-intercept 3 and y-intercept 6.
Show Answer

The intercepts are (3, 0) and (0, 6).

m = y₂ − y₁x₂ − x₁

m = −63

m = −2

The y-intercept is (0, 6).

y = −2x + 6

Add 2x to both sides.

2x + y = 6

Answer: 2x + y = 6

3
Write Ax + By = C for the line with x-intercept 2 and y-intercept −4.
Show Answer

The intercepts are (2, 0) and (0, −4).

m = y₂ − y₁x₂ − x₁

m = 42

m = 2

The y-intercept is (0, −4).

y = 2x − 4

Subtract 2x from both sides.

−2x + y = −4

A is negative. Multiply by −1.

2x − y = 4

Answer: 2x − y = 4

4
Write Ax + By = C for the line with x-intercept −2 and y-intercept 6.
Show Answer

The intercepts are (−2, 0) and (0, 6).

m = y₂ − y₁x₂ − x₁

m = −6−2

m = 3

The y-intercept is (0, 6).

y = 3x + 6

Subtract 3x from both sides.

−3x + y = 6

A is negative. Multiply by −1.

3x − y = −6

Answer: 3x − y = −6

5
Write Ax + By = C for the line with x-intercept 8 and y-intercept 2.
Show Answer

The intercepts are (8, 0) and (0, 2).

m = y₂ − y₁x₂ − x₁

m = −28

m = −14

The y-intercept is (0, 2).

y = −14x + 2

Add 14x to both sides.

14x + y = 2

Multiply by 4.

x + 4y = 8

Answer: x + 4y = 8

Medium

Slope and a point

Write point-slope form, then standard form.

Example
Write Ax + By = C for the line with slope 2 through (1, 5).

y − y₁ = m(x − x₁)

y − 5 = 2(x − 1)

Distribute.

y − 5 = 2x − 2

Add 5 to both sides.

y = 2x + 3

Subtract 2x from both sides.

−2x + y = 3

A is negative. Multiply by −1.

2x − y = −3

Answer: 2x − y = −3

1
Write Ax + By = C for the line with slope −3 through (2, 1).
Show Answer

y − y₁ = m(x − x₁)

y − 1 = −3(x − 2)

Distribute.

y − 1 = −3x + 6

Add 1 to both sides.

y = −3x + 7

Add 3x to both sides.

3x + y = 7

Answer: 3x + y = 7

2
Write Ax + By = C for the line with slope 12 through (4, 3).
Show Answer

y − y₁ = m(x − x₁)

y − 3 = 12(x − 4)

Distribute.

y − 3 = 12x − 2

Add 3 to both sides.

y = 12x + 1

Subtract 12x from both sides.

−12x + y = 1

Multiply by 2.

−x + 2y = 2

A is negative. Multiply by −1.

x − 2y = −2

Answer: x − 2y = −2

3
Write Ax + By = C for the line with slope −23 through (3, 4).
Show Answer

y − y₁ = m(x − x₁)

y − 4 = −23(x − 3)

Distribute.

y − 4 = −23x + 2

Add 4 to both sides.

y = −23x + 6

Add 23x to both sides.

23x + y = 6

Multiply by 3.

2x + 3y = 18

Answer: 2x + 3y = 18

4
Write Ax + By = C for the line with slope 4 through (−1, 2).
Show Answer

y − y₁ = m(x − x₁)

y − 2 = 4(x + 1)

Distribute.

y − 2 = 4x + 4

Add 2 to both sides.

y = 4x + 6

Subtract 4x from both sides.

−4x + y = 6

A is negative. Multiply by −1.

4x − y = −6

Answer: 4x − y = −6

5
Write Ax + By = C for the line with slope −14 through (8, 1).
Show Answer

y − y₁ = m(x − x₁)

y − 1 = −14(x − 8)

Distribute.

y − 1 = −14x + 2

Add 1 to both sides.

y = −14x + 3

Add 14x to both sides.

14x + y = 3

Multiply by 4.

x + 4y = 12

Answer: x + 4y = 12

Hard

Two points

Find the slope from the two points, then write standard form.

Example
Write an equation of the line through (1, 2) and (3, 8) in standard form.

m = y₂ − y₁x₂ − x₁

m = 62

m = 3

Use (1, 2).

y = mx + b

2 = 3(1) + b

2 = 3 + b

2 − 3 = b

b = −1

y = 3x − 1

Subtract 3x from both sides.

−3x + y = −1

A is negative. Multiply by −1.

3x − y = 1

Answer: 3x − y = 1

1
Write an equation of the line through (0, 4) and (2, 0) in standard form.
Show Answer

m = y₂ − y₁x₂ − x₁

m = −42

m = −2

Use (0, 4).

y = mx + b

4 = −2(0) + b

4 = 0 + b

4 − 0 = b

b = 4

y = −2x + 4

Add 2x to both sides.

2x + y = 4

Answer: 2x + y = 4

2
Write an equation of the line through (1, 5) and (4, −1) in standard form.
Show Answer

m = y₂ − y₁x₂ − x₁

m = −63

m = −2

Use (1, 5).

y = mx + b

5 = −2(1) + b

5 = −2 + b

5 + 2 = b

b = 7

y = −2x + 7

Add 2x to both sides.

2x + y = 7

Answer: 2x + y = 7

3
Write an equation of the line through (−2, 3) and (2, 5) in standard form.
Show Answer

m = y₂ − y₁x₂ − x₁

m = 24

m = 12

Use (−2, 3).

y = mx + b

3 = 12(−2) + b

3 = −1 + b

3 + 1 = b

b = 4

y = 12x + 4

Subtract 12x from both sides.

−12x + y = 4

Multiply by 2.

−x + 2y = 8

A is negative. Multiply by −1.

x − 2y = −8

Answer: x − 2y = −8

4
Write an equation of the line through (5, 1) and (5, 7) in standard form.
Show Answer

The x-values are the same, so the run is 0.

The slope is undefined.

A vertical line is x = 5.

Answer: x = 5

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

m = y₂ − y₁x₂ − x₁

m = 84

m = 2

Use (2, −3).

y = mx + b

−3 = 2(2) + b

−3 = 4 + b

−3 − 4 = b

b = −7

y = 2x − 7

Subtract 2x from both sides.

−2x + y = −7

A is negative. Multiply by −1.

2x − y = 7

Answer: 2x − y = 7

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