Write y − y₁ = m(x − x₁)

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Key Ideas Easy Medium Hard

What to Remember

Point-slope form

y − y₁ = m(x − x₁). Put the slope in for m and the point in for (x₁, y₁).

The signs

A positive x₁ or y₁ stays after a minus. A negative x₁ or y₁ becomes a plus.

Two points

Find the slope, then use the first point. If the x-values match, the line is vertical: x = a.

Easy

Integer slope

The slope and the point are given.

Example

Write the point-slope equation with slope 2 through (3, 5).

y − y₁ = m(x − x₁)

m = 2

The point is (3, 5).

y − 5 = 2(x − 3)

Answer: y − 5 = 2(x − 3)

Write the point-slope equation with slope −4 through (1, 6).

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y − y₁ = m(x − x₁)

m = −4

The point is (1, 6).

y − 6 = −4(x − 1)

Answer: y − 6 = −4(x − 1)

Write the point-slope equation with slope 3 through (−2, 4).

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y − y₁ = m(x − x₁)

m = 3

The point is (−2, 4).

y − 4 = 3(x + 2)

Answer: y − 4 = 3(x + 2)

Write the point-slope equation with slope −1 through (5, −3).

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y − y₁ = m(x − x₁)

m = −1

The point is (5, −3).

y + 3 = −(x − 5)

Answer: y + 3 = −(x − 5)

Write the point-slope equation with slope 5 through (2, 0).

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y − y₁ = m(x − x₁)

m = 5

The point is (2, 0).

y = 5(x − 2)

Answer: y = 5(x − 2)

Write the point-slope equation with slope −2 through (8, 1).

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y − y₁ = m(x − x₁)

m = −2

The point is (8, 1).

y − 1 = −2(x − 8)

Answer: y − 1 = −2(x − 8)

Medium

Fractional slope

The slope is a positive or negative fraction.

Example

Write the point-slope equation with slope 12 through (4, 3).

y − y₁ = m(x − x₁)

m = 12

The point is (4, 3).

y − 3 = 12(x − 4)

Answer: y − 3 = 12(x − 4)

Write the point-slope equation with slope −34 through (−4, 1).

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y − y₁ = m(x − x₁)

m = −34

The point is (−4, 1).

y − 1 = −34(x + 4)

Answer: y − 1 = −34(x + 4)

Write the point-slope equation with slope 25 through (5, −2).

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y − y₁ = m(x − x₁)

m = 25

The point is (5, −2).

y + 2 = 25(x − 5)

Answer: y + 2 = 25(x − 5)

Write the point-slope equation with slope −13 through (6, 4).

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y − y₁ = m(x − x₁)

m = −13

The point is (6, 4).

y − 4 = −13(x − 6)

Answer: y − 4 = −13(x − 6)

Write the point-slope equation with slope 32 through (−2, −5).

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y − y₁ = m(x − x₁)

m = 32

The point is (−2, −5).

y + 5 = 32(x + 2)

Answer: y + 5 = 32(x + 2)

Write the point-slope equation with slope 14 through (−8, 2).

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y − y₁ = m(x − x₁)

m = 14

The point is (−8, 2).

y − 2 = 14(x + 8)

Answer: y − 2 = 14(x + 8)

Hard

Two points

Find the slope, then use the first point.

Example

Write the point-slope equation of the line through (1, 2) and (4, 8). Use (1, 2).

m = y₂ − y₁x₂ − x₁

m = 63

m = 2

Use (1, 2).

y − y₁ = m(x − x₁)

y − 2 = 2(x − 1)

Answer: y − 2 = 2(x − 1)

Write the point-slope equation of the line through (2, 5) and (6, 13). Use (2, 5).

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m = y₂ − y₁x₂ − x₁

m = 84

m = 2

Use (2, 5).

y − y₁ = m(x − x₁)

y − 5 = 2(x − 2)

Answer: y − 5 = 2(x − 2)

Write the point-slope equation of the line through (−1, 4) and (3, −4). Use (−1, 4).

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m = y₂ − y₁x₂ − x₁

m = −84

m = −2

Use (−1, 4).

y − y₁ = m(x − x₁)

y − 4 = −2(x + 1)

Answer: y − 4 = −2(x + 1)

Write the point-slope equation of the line through (2, 1) and (6, 3). Use (2, 1).

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m = y₂ − y₁x₂ − x₁

m = 24

m = 12

Use (2, 1).

y − y₁ = m(x − x₁)

y − 1 = 12(x − 2)

Answer: y − 1 = 12(x − 2)

Write the point-slope equation of the line through (3, 7) and (3, −1). Use (3, 7).

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

The slope is undefined, so point-slope form does not apply.

The equation is x = 3.

Answer: x = 3

Write the point-slope equation of the line through (4, −2) and (8, 4). Use (4, −2).

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m = y₂ − y₁x₂ − x₁

m = 64

m = 32

Use (4, −2).

y − y₁ = m(x − x₁)

y + 2 = 32(x − 4)

Answer: y + 2 = 32(x − 4)

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