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Amy Ferguson Moncure

IB AA · 4A Equations of Straight Lines

Equation through Points

Build a line from a gradient and a point, or from two points.

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Key Ideas Easy Medium Hard

y − y₁ = m(x − x₁)

What to Remember

Point-gradient form

y − y₁ = m(x − x₁)

Then expand and solve for y.

Two points

Find the gradient first. Then use either point in the point-gradient form.

Standard form

ax + by + c = 0 is the same line with every term on one side.

Vertical

If the x-coordinates match, the equation is x = that number.

Easy

Easy

You are given the gradient and one point.

Example
Gradient 2 through (3, 1). Write y = mx + c.

y − y₁ = m(x − x₁)

y − 1 = 2(x − 3)

y − 1 = 2x − 6

y = 2x − 5

Answer: y = 2x − 5

1
Find the equation of the line with gradient 2 through (1, 5). Give the answer as y = mx + c.
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y − y₁ = m(x − x₁)

y − 5 = 2(x − 1)

y − 5 = 2x − 2

y = 2x + 3

Answer: y = 2x + 3

2
Find the equation of the line with gradient −1 through (3, 4). Give the answer as y = mx + c.
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y − y₁ = m(x − x₁)

y − 4 = −1(x − 3)

y − 4 = −x + 3

y = −x + 7

Answer: y = −x + 7

3
Find the equation of the line with gradient 3 through (0, −2). Give the answer as y = mx + c.
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y − y₁ = m(x − x₁)

y − (−2) = 3(x − 0)

y − (−2) = 3x

y = 3x − 2

Answer: y = 3x − 2

4
Find the equation of the line with gradient 0 through (4, 7). Give the answer as y = mx + c.
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y − y₁ = m(x − x₁)

y − 7 = 0(x − 4)

y − 7 = 0

y = 7

Answer: y = 7

5
Find the equation of the line with gradient 4 through (−1, 2). Give the answer as y = mx + c.
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y − y₁ = m(x − x₁)

y − 2 = 4(x − (−1))

y − 2 = 4x + 4

y = 4x + 6

Answer: y = 4x + 6

Medium

Medium

Two points. Write y = mx + c.

Example
Through (1, 4) and (3, 10). Write y = mx + c.

m = y₂ − y₁x₂ − x₁

m = 10 − 43 − 1

m = 62

m = 3

y − y₁ = m(x − x₁)

y − 4 = 3(x − 1)

y − 4 = 3x − 3

y = 3x + 1

Answer: y = 3x + 1

1
Find the equation of the line through (1, 2) and (3, 8). Give the answer as y = mx + c.
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m = y₂ − y₁x₂ − x₁

m = 8 − 23 − 1

m = 62

m = 3

y − y₁ = m(x − x₁)

y − 2 = 3(x − 1)

y − 2 = 3x − 3

y = 3x − 1

Answer: y = 3x − 1

2
Find the equation of the line through (0, 5) and (4, −3). Give the answer as y = mx + c.
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m = y₂ − y₁x₂ − x₁

m = −3 − 54 − 0

m = −84

m = −2

y − y₁ = m(x − x₁)

y − 5 = −2(x − 0)

y − 5 = −2x

y = −2x + 5

Answer: y = −2x + 5

3
Find the equation of the line through (−2, 1) and (2, 9). Give the answer as y = mx + c.
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m = y₂ − y₁x₂ − x₁

m = 9 − 12 − (−2)

m = 84

m = 2

y − y₁ = m(x − x₁)

y − 1 = 2(x − (−2))

y − 1 = 2x + 4

y = 2x + 5

Answer: y = 2x + 5

4
Find the equation of the line through (3, 3) and (6, 3). Give the answer as y = mx + c.
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m = y₂ − y₁x₂ − x₁

m = 3 − 36 − 3

m = 03

m = 0

y − y₁ = m(x − x₁)

y − 3 = 0(x − 3)

y − 3 = 0

y = 3

Answer: y = 3

5
Find the equation of the line through (1, −4) and (5, 4). Give the answer as y = mx + c.
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m = y₂ − y₁x₂ − x₁

m = 4 − (−4)5 − 1

m = 84

m = 2

y − y₁ = m(x − x₁)

y − (−4) = 2(x − 1)

y − (−4) = 2x − 2

y = 2x − 6

Answer: y = 2x − 6

Hard

Hard

Two points. Write ax + by + c = 0.

Example
Through (0, −2) and (3, 4). Write ax + by + c = 0.

m = y₂ − y₁x₂ − x₁

m = 4 − (−2)3 − 0

m = 63

m = 2

y − y₁ = m(x − x₁)

y − (−2) = 2(x − 0)

y − (−2) = 2x

y = 2x − 2

2x − y − 2 = 0

Answer: 2x − y − 2 = 0

1
Find the equation of the line through (2, 1) and (6, 3). Give the answer as ax + by + c = 0.
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m = y₂ − y₁x₂ − x₁

m = 3 − 16 − 2

m = 24

m = 12

y − y₁ = m(x − x₁)

y − 1 = 12(x − 2)

y − 1 = 12x − 1

y = 12x

x − 2y = 0

Answer: x − 2y = 0

2
Find the equation of the line through (−1, 4) and (3, −2). Give the answer as ax + by + c = 0.
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m = y₂ − y₁x₂ − x₁

m = −2 − 43 − (−1)

m = −64

m = −32

y − y₁ = m(x − x₁)

y − 4 = −32(x − (−1))

y − 4 = −32x − 32

y = −32x + 52

3x + 2y − 5 = 0

Answer: 3x + 2y − 5 = 0

3
Find the equation of the line through (0, 2) and (4, −4). Give the answer as ax + by + c = 0.
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m = y₂ − y₁x₂ − x₁

m = −4 − 24 − 0

m = −64

m = −32

y − y₁ = m(x − x₁)

y − 2 = −32(x − 0)

y − 2 = −32x

y = −32x + 2

3x + 2y − 4 = 0

Answer: 3x + 2y − 4 = 0

4
Find the equation of the line through (5, 1) and (5, 6). Give the answer as ax + by + c = 0.
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m = y₂ − y₁x₂ − x₁

m = 6 − 15 − 5

The run is 0, so the line is vertical.

x = 5

Answer: x = 5

5
Find the equation of the line through (−3, −1) and (1, 5). Give the answer as ax + by + c = 0.
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m = y₂ − y₁x₂ − x₁

m = 5 − (−1)1 − (−3)

m = 64

m = 32

y − y₁ = m(x − x₁)

y − (−1) = 32(x − (−3))

y − (−1) = 32x + 92

y = 32x + 72

3x − 2y + 7 = 0

Answer: 3x − 2y + 7 = 0

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