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


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

8x + 40 -16x = -40

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


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

8x + 40 -16x = -40

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

-8x + 40 = -40


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

8x + 40 -16x = -40

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

-8x + 40 = -40

-8x = -80


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

8x + 40 -16x = -40

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

-8x + 40 = -40

-8x = -80

x = 10


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

8x + 40 -16x = -40

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

-8x + 40 = -40

-8x = -80

x = 10


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


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


3(4y + 1) - 2

12y + 3 - 2

12y + 1

1. Tables

1. Tables Not very efficient


2. Graphing by hand


2. Graphing by hand Not very accurate



3. Graphing Calculator



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.




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





⎧

⎨⎪

⎩⎪





⎧

⎨⎪

⎩⎪

A + S + C = 1750





⎧

⎨⎪

⎩⎪

A + S + C = 1750

S = 2A





⎧

⎨⎪

⎩⎪

A + S + C = 1750

S = 2A

C = 1/2 A





⎧

⎨⎪

⎩⎪

A + S + C = 1750

S = 2A

C = 1/2 A

A + 2A + 1/2 A = 1750





⎧

⎨⎪

⎩⎪

A + S + C = 1750

S = 2A

C = 1/2 A

A + 2A + 1/2 A = 1750

7/2 A = 1750





⎧

⎨⎪

⎩⎪

A + S + C = 1750

S = 2A

C = 1/2 A

A + 2A + 1/2 A = 1750

7/2 A = 1750

A = 500





⎧

⎨⎪

⎩⎪

A + S + C = 1750

S = 2A

C = 1/2 A

A + 2A + 1/2 A = 1750

7/2 A = 1750

A = 500

S = 1000





⎧

⎨⎪

⎩⎪

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





⎧

⎨⎪

⎩⎪

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


These are the same graphs.

Example 5

y = 2x2

3y = 6x2

⎧⎨⎪

⎩⎪

Solve.

3(2x2 ) = 6x2

6x2 = 6x2


These are the same graphs.

Infinitely many solutions on the parabola

Consistent:

Consistent: A system with one or more solutions


Inconsistent:


Inconsistent: A systems with no solutions

Homework

Homework

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

“Too many people are thinking of security instead of opportunity. They seem more afraid of life than death.” - James F. Byrnes

AA Section 5-3

Entertainment & Humor

Transcript of AA Section 5-3