AA Section 5-3

Section 5-3Solving Systems by Substitution

Warm-up1. Solve 8x + 8(5-2x) = -40

2. Evaluate 3x - 2 when x = 4y + 1

Warm-up1. Solve 8x + 8(5-2x) = -40

8x + 8(5-2x) = -40

2. Evaluate 3x - 2 when x = 4y + 1

Warm-up1. Solve 8x + 8(5-2x) = -40

8x + 40 -16x = -40

8x + 8(5-2x) = -40

2. Evaluate 3x - 2 when x = 4y + 1

Warm-up1. Solve 8x + 8(5-2x) = -40

8x + 40 -16x = -40

8x + 8(5-2x) = -40

-8x + 40 = -40

2. Evaluate 3x - 2 when x = 4y + 1

Warm-up1. Solve 8x + 8(5-2x) = -40

8x + 40 -16x = -40

8x + 8(5-2x) = -40

-8x + 40 = -40

-8x = -80

2. Evaluate 3x - 2 when x = 4y + 1

Warm-up1. Solve 8x + 8(5-2x) = -40

8x + 40 -16x = -40

8x + 8(5-2x) = -40

-8x + 40 = -40

-8x = -80

x = 10

2. Evaluate 3x - 2 when x = 4y + 1

Warm-up1. Solve 8x + 8(5-2x) = -40

8x + 40 -16x = -40

8x + 8(5-2x) = -40

-8x + 40 = -40

-8x = -80

x = 10

2. Evaluate 3x - 2 when x = 4y + 1

3(4y + 1) - 2

Warm-up1. Solve 8x + 8(5-2x) = -40

8x + 40 -16x = -40

8x + 8(5-2x) = -40

-8x + 40 = -40

-8x = -80

x = 10

2. Evaluate 3x - 2 when x = 4y + 1

3(4y + 1) - 2

12y + 3 - 2

Warm-up1. Solve 8x + 8(5-2x) = -40

8x + 40 -16x = -40

8x + 8(5-2x) = -40

-8x + 40 = -40

-8x = -80

x = 10

2. Evaluate 3x - 2 when x = 4y + 1

3(4y + 1) - 2

12y + 3 - 2

12y + 1

1. Tables

1. Tables Not very efficient

1. Tables Not very efficient

2. Graphing by hand

1. Tables Not very efficient

2. Graphing by hand Not very accurate

1. Tables Not very efficient

2. Graphing by hand Not very accurate

3. Graphing Calculator

1. Tables Not very efficient

2. Graphing by hand Not very accurate

3. Graphing Calculator Cheap way out

Example 1Solve.

x + y = 6

y = x + 2

⎧⎨⎩⎪

Example 1Solve.

x + y = 6

y = x + 2

⎧⎨⎩⎪

x + (x + 2) = 6

Example 1Solve.

x + y = 6

y = x + 2

⎧⎨⎩⎪

x + (x + 2) = 6

2x + 2 = 6

Example 1Solve.

x + y = 6

y = x + 2

⎧⎨⎩⎪

x + (x + 2) = 6

2x + 2 = 6

2x = 4

Example 1Solve.

x + y = 6

y = x + 2

⎧⎨⎩⎪

x + (x + 2) = 6

2x + 2 = 6

2x = 4

x = 2

Example 1Solve.

x + y = 6

y = x + 2

⎧⎨⎩⎪

x + (x + 2) = 6

2x + 2 = 6

2x = 4

x = 2

y = x + 2

Example 1Solve.

x + y = 6

y = x + 2

⎧⎨⎩⎪

x + (x + 2) = 6

2x + 2 = 6

2x = 4

x = 2

y = x + 2

y = 2 + 2

Example 1Solve.

x + y = 6

y = x + 2

⎧⎨⎩⎪

x + (x + 2) = 6

2x + 2 = 6

2x = 4

x = 2

y = x + 2

y = 2 + 2

y = 4

Example 1Solve.

x + y = 6

y = x + 2

⎧⎨⎩⎪

x + (x + 2) = 6

2x + 2 = 6

2x = 4

x = 2

y = x + 2

y = 2 + 2

y = 4

x + y = 6

Example 1Solve.

x + y = 6

y = x + 2

⎧⎨⎩⎪

x + (x + 2) = 6

2x + 2 = 6

2x = 4

x = 2

y = x + 2

y = 2 + 2

y = 4

x + y = 6

2 + 4 = 6

Example 1Solve.

x + y = 6

y = x + 2

⎧⎨⎩⎪

x + (x + 2) = 6

2x + 2 = 6

2x = 4

x = 2

y = x + 2

y = 2 + 2

y = 4

x + y = 6

2 + 4 = 6

(2, 4)

Example 1Solve.

x + y = 6

y = x + 2

⎧⎨⎩⎪

x + (x + 2) = 6

2x + 2 = 6

2x = 4

x = 2

y = x + 2

y = 2 + 2

y = 4

x + y = 6

2 + 4 = 6

(2, 4)

Always check your answer.

Example 1Solve.

x + y = 6

y = x + 2

⎧⎨⎩⎪

x + (x + 2) = 6

2x + 2 = 6

2x = 4

x = 2

y = x + 2

y = 2 + 2

y = 4

x + y = 6

2 + 4 = 6

(2, 4)

Always check your answer.

You’ll know you’re right.

Example 2The Drama Club printed 1750 tickets for their spring play. They

printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and

find the number of each ticket printed.

Example 2The Drama Club printed 1750 tickets for their spring play. They

printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and

find the number of each ticket printed.

A = adult tickets S = student tickets C = children tickets

Example 2The Drama Club printed 1750 tickets for their spring play. They

printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and

find the number of each ticket printed.

A = adult tickets S = student tickets C = children tickets

⎧

⎨⎪

⎩⎪

Example 2The Drama Club printed 1750 tickets for their spring play. They

printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and

find the number of each ticket printed.

A = adult tickets S = student tickets C = children tickets

⎧

⎨⎪

⎩⎪

A + S + C = 1750

Example 2The Drama Club printed 1750 tickets for their spring play. They

printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and

find the number of each ticket printed.

A = adult tickets S = student tickets C = children tickets

⎧

⎨⎪

⎩⎪

A + S + C = 1750

S = 2A

Example 2The Drama Club printed 1750 tickets for their spring play. They

printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and

find the number of each ticket printed.

A = adult tickets S = student tickets C = children tickets

⎧

⎨⎪

⎩⎪

A + S + C = 1750

S = 2A

C = 1/2 A

Example 2The Drama Club printed 1750 tickets for their spring play. They

printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and

find the number of each ticket printed.

A = adult tickets S = student tickets C = children tickets

⎧

⎨⎪

⎩⎪

A + S + C = 1750

S = 2A

C = 1/2 A

A + 2A + 1/2 A = 1750

Example 2The Drama Club printed 1750 tickets for their spring play. They

printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and

find the number of each ticket printed.

A = adult tickets S = student tickets C = children tickets

⎧

⎨⎪

⎩⎪

A + S + C = 1750

S = 2A

C = 1/2 A

A + 2A + 1/2 A = 1750

7/2 A = 1750

Example 2The Drama Club printed 1750 tickets for their spring play. They

printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and

find the number of each ticket printed.

A = adult tickets S = student tickets C = children tickets

⎧

⎨⎪

⎩⎪

A + S + C = 1750

S = 2A

C = 1/2 A

A + 2A + 1/2 A = 1750

7/2 A = 1750

A = 500

Example 2The Drama Club printed 1750 tickets for their spring play. They

printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and

find the number of each ticket printed.

A = adult tickets S = student tickets C = children tickets

⎧

⎨⎪

⎩⎪

A + S + C = 1750

S = 2A

C = 1/2 A

A + 2A + 1/2 A = 1750

7/2 A = 1750

A = 500

S = 1000

Example 2The Drama Club printed 1750 tickets for their spring play. They

printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and

find the number of each ticket printed.

A = adult tickets S = student tickets C = children tickets

⎧

⎨⎪

⎩⎪

A + S + C = 1750

S = 2A

C = 1/2 A

A + 2A + 1/2 A = 1750

7/2 A = 1750

A = 500

S = 1000

C = 250

Example 2The Drama Club printed 1750 tickets for their spring play. They

printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and

find the number of each ticket printed.

A = adult tickets S = student tickets C = children tickets

⎧

⎨⎪

⎩⎪

A + S + C = 1750

S = 2A

C = 1/2 A

A + 2A + 1/2 A = 1750

7/2 A = 1750

A = 500

S = 1000

C = 250

They printed 500 adult tickets, 1000 student tickets, and 250 children’s tickets

Example 3Solve.

y = 4x

xy = 36

⎧⎨⎩⎪

Example 3Solve.

y = 4x

xy = 36

⎧⎨⎩⎪

x(4x) = 36

Example 3Solve.

y = 4x

xy = 36

⎧⎨⎩⎪

x(4x) = 36

4x2 = 36

Example 3Solve.

y = 4x

xy = 36

⎧⎨⎩⎪

x(4x) = 36

4x2 = 36

x2 = 9

Example 3Solve.

y = 4x

xy = 36

⎧⎨⎩⎪

x(4x) = 36

4x2 = 36

x2 = 9

x2 = ± 9

Example 3Solve.

y = 4x

xy = 36

⎧⎨⎩⎪

x(4x) = 36

4x2 = 36

x2 = 9

x = 3 or x = -3 x2 = ± 9

Example 3Solve.

y = 4x

xy = 36

⎧⎨⎩⎪

x(4x) = 36

4x2 = 36

x2 = 9

x = 3 or x = -3

y = 4(3)

x2 = ± 9

Example 3Solve.

y = 4x

xy = 36

⎧⎨⎩⎪

x(4x) = 36

4x2 = 36

x2 = 9

x = 3 or x = -3

y = 4(3)

y = 12

x2 = ± 9

Example 3Solve.

y = 4x

xy = 36

⎧⎨⎩⎪

x(4x) = 36

4x2 = 36

x2 = 9

x = 3 or x = -3

y = 4(3)

y = 12

y = 4(-3)

x2 = ± 9

Example 3Solve.

y = 4x

xy = 36

⎧⎨⎩⎪

x(4x) = 36

4x2 = 36

x2 = 9

x = 3 or x = -3

y = 4(3)

y = 12

y = -12

y = 4(-3)

x2 = ± 9

Example 3Solve.

y = 4x

xy = 36

⎧⎨⎩⎪

x(4x) = 36

4x2 = 36

x2 = 9

x = 3 or x = -3

y = 4(3)

y = 12

y = -12

y = 4(-3)

Check:

x2 = ± 9

Example 3Solve.

y = 4x

xy = 36

⎧⎨⎩⎪

x(4x) = 36

4x2 = 36

x2 = 9

x = 3 or x = -3

y = 4(3)

y = 12

y = -12

y = 4(-3)

Check:

(3)(12) = 36

x2 = ± 9

Example 3Solve.

y = 4x

xy = 36

⎧⎨⎩⎪

x(4x) = 36

4x2 = 36

x2 = 9

x = 3 or x = -3

y = 4(3)

y = 12

y = -12

y = 4(-3)

Check:

(3)(12) = 36

(-3)(-12) = 36

x2 = ± 9

Example 3Solve.

y = 4x

xy = 36

⎧⎨⎩⎪

x(4x) = 36

4x2 = 36

x2 = 9

x = 3 or x = -3

y = 4(3)

y = 12

y = -12

y = 4(-3)

Check:

(3)(12) = 36

(-3)(-12) = 36

(3, 12) or (-3, -12)

x2 = ± 9

Example 4Solve.

y = 4 − 3x

3x + y = 7

⎧⎨⎩⎪

Example 4Solve.

y = 4 − 3x

3x + y = 7

⎧⎨⎩⎪

3x + (4 - 3x) = 7

Example 4Solve.

y = 4 − 3x

3x + y = 7

⎧⎨⎩⎪

3x + (4 - 3x) = 74 = 7

Example 4Solve.

y = 4 − 3x

3x + y = 7

⎧⎨⎩⎪

3x + (4 - 3x) = 74 ≠ 7

Example 4Solve.

y = 4 − 3x

3x + y = 7

⎧⎨⎩⎪

3x + (4 - 3x) = 7

Wait, what?

4 ≠ 7

Example 4Solve.

y = 4 − 3x

3x + y = 7

⎧⎨⎩⎪

3x + (4 - 3x) = 7

Wait, what?

3x + y = 7

4 ≠ 7

Example 4Solve.

y = 4 − 3x

3x + y = 7

⎧⎨⎩⎪

3x + (4 - 3x) = 7

Wait, what?

3x + y = 7

y = -3x + 7

4 ≠ 7

Example 4Solve.

y = 4 − 3x

3x + y = 7

⎧⎨⎩⎪

3x + (4 - 3x) = 7

Wait, what?

3x + y = 7

y = -3x + 7

Oh, parallel lines!

4 ≠ 7

Example 4Solve.

y = 4 − 3x

3x + y = 7

⎧⎨⎩⎪

3x + (4 - 3x) = 7

Wait, what?

3x + y = 7

y = -3x + 7

Oh, parallel lines!

4 ≠ 7

(No solutions)

Example 5

y = 2x2

3y = 6x2

⎧⎨⎪

⎩⎪

Solve.

Example 5

y = 2x2

3y = 6x2

⎧⎨⎪

⎩⎪

Solve.

3(2x2 ) = 6x2

Example 5

y = 2x2

3y = 6x2

⎧⎨⎪

⎩⎪

Solve.

3(2x2 ) = 6x2

6x2 = 6x2

Example 5

y = 2x2

3y = 6x2

⎧⎨⎪

⎩⎪

Solve.

3(2x2 ) = 6x2

6x2 = 6x2

This is always true!

Example 5

y = 2x2

3y = 6x2

⎧⎨⎪

⎩⎪

Solve.

3(2x2 ) = 6x2

6x2 = 6x2

This is always true!

These are the same graphs.

Example 5

y = 2x2

3y = 6x2

⎧⎨⎪

⎩⎪

Solve.

3(2x2 ) = 6x2

6x2 = 6x2

This is always true!

These are the same graphs.

Infinitely many solutions on the parabola

Consistent:

Consistent: A system with one or more solutions

Consistent: A system with one or more solutions

Inconsistent:

Consistent: A system with one or more solutions

Inconsistent: A systems with no solutions

Homework

Homework

p. 289 #1-20, skip 17, 18

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