Point of Intersection | Math Coach Amy

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Key Ideas Easy Medium Hard

Set the two right sides equal.

What to Remember

Same y

At the intersection, the two expressions for y are equal.

Solve for x

Collect the x terms on one side and the constants on the other.

Then y

Substitute that x into either line.

Check

The point should satisfy both equations.

Easy

Integer gradients.

Example

Find the point where y = 2x − 1 meets y = −x + 8.

y = 2x − 1

y = −x + 8

Set the right sides equal.

2x − 1 = −x + 8

3x − 9 = 0

3x = 9

x = 3

y = 2(3) − 1

y = 5

Answer: (3, 5)

Find the point where y = 2x + 1 meets y = −x + 7.

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y = 2x + 1

y = −x + 7

Set the right sides equal.

2x + 1 = −x + 7

3x − 6 = 0

3x = 6

x = 2

y = 2(2) + 1

y = 5

Answer: (2, 5)

Find the point where y = x + 4 meets y = 3x − 2.

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y = x + 4

y = 3x − 2

Set the right sides equal.

x + 4 = 3x − 2

−2x + 6 = 0

−2x = −6

x = 3

y = 3 + 4

y = 7

Answer: (3, 7)

Find the point where y = −2x + 8 meets y = x − 1.

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y = −2x + 8

y = x − 1

Set the right sides equal.

−2x + 8 = x − 1

−3x + 9 = 0

−3x = −9

x = 3

y = −2(3) + 8

y = 2

Answer: (3, 2)

Find the point where y = 3x − 6 meets y = x + 2.

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y = 3x − 6

y = x + 2

Set the right sides equal.

3x − 6 = x + 2

2x − 8 = 0

2x = 8

x = 4

y = 3(4) − 6

y = 6

Answer: (4, 6)

Find the point where y = 12x + 1 meets y = 2x − 2.

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y = 12x + 1

y = 2x − 2

Set the right sides equal.

12x + 1 = 2x − 2

−32x + 3 = 0

−32x = −3

x = 2

y = 12(2) + 1

y = 2

Answer: (2, 2)

Medium

One gradient is a fraction, or the numbers are larger.

Example

Find the point where y = 4x − 3 meets y = −x + 7.

y = 4x − 3

y = −x + 7

Set the right sides equal.

4x − 3 = −x + 7

5x − 10 = 0

5x = 10

x = 2

y = 4(2) − 3

y = 5

Answer: (2, 5)

Find the point where y = 23x + 1 meets y = −13x + 4.

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y = 23x + 1

y = −13x + 4

Set the right sides equal.

23x + 1 = −13x + 4

x − 3 = 0

x = 3

y = 23(3) + 1

y = 3

Answer: (3, 3)

Find the point where y = 5x + 1 meets y = 2x + 10.

Show Answer

y = 5x + 1

y = 2x + 10

Set the right sides equal.

5x + 1 = 2x + 10

3x − 9 = 0

3x = 9

x = 3

y = 5(3) + 1

y = 16

Answer: (3, 16)

Find the point where y = −3x + 4 meets y = 2x − 6.

Show Answer

y = −3x + 4

y = 2x − 6

Set the right sides equal.

−3x + 4 = 2x − 6

−5x + 10 = 0

−5x = −10

x = 2

y = −3(2) + 4

y = −2

Answer: (2, −2)

Find the point where y = x − 5 meets y = −2x + 4.

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y = x − 5

y = −2x + 4

Set the right sides equal.

x − 5 = −2x + 4

3x − 9 = 0

3x = 9

x = 3

y = 3 − 5

y = −2

Answer: (3, −2)

Find the point where y = 3x + 2 meets y = −12x + 9.

Show Answer

y = 3x + 2

y = −12x + 9

Set the right sides equal.

3x + 2 = −12x + 9

72x − 7 = 0

72x = 7

x = 2

y = 3(2) + 2

y = 8

Answer: (2, 8)

Hard

Both lines can have a fraction, or one line is horizontal.

Example

Find the point where y = 6x − 8 meets y = x + 2.

y = 6x − 8

y = x + 2

Set the right sides equal.

6x − 8 = x + 2

5x − 10 = 0

5x = 10

x = 2

y = 6(2) − 8

y = 4

Answer: (2, 4)

Find the point where y = 32x − 1 meets y = 12x + 3.

Show Answer

y = 32x − 1

y = 12x + 3

Set the right sides equal.

32x − 1 = 12x + 3

x − 4 = 0

x = 4

y = 32(4) − 1

y = 5

Answer: (4, 5)

Find the point where y = −4x + 1 meets y = 2x + 13.

Show Answer

y = −4x + 1

y = 2x + 13

Set the right sides equal.

−4x + 1 = 2x + 13

−6x − 12 = 0

−6x = 12

x = −2

y = −4(−2) + 1

y = 9

Answer: (−2, 9)

Find the point where y = 3x − 9 meets y = 3.

Show Answer

y = 3x − 9

y = 3

Set the right sides equal.

3x − 9 = 3

3x − 12 = 0

3x = 12

x = 4

y = 3(4) − 9

y = 3

Answer: (4, 3)

Find the point where y = −34x + 6 meets y = 14x + 2.

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y = −34x + 6

y = 14x + 2

Set the right sides equal.

−34x + 6 = 14x + 2

−x + 4 = 0

−x = −4

x = 4

y = −34(4) + 6

y = 3

Answer: (4, 3)

Find the point where y = 2x − 4 meets y = −23x + 6.

Show Answer

y = 2x − 4

y = −23x + 6

Set the right sides equal.

2x − 4 = −23x + 6

83x − 10 = 0

83x = 10

x = 154

y = 2(154) − 4

y = 72

Answer: (154, 72)

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