Systems with Three Variables

Objective:

I can solve systems with three or more variables.

How much does each box weigh?Explain your reasoning.

{ 𝟐 𝒙 −𝒚+𝒛=𝟒𝒙+𝟑 𝒚− 𝒛=𝟏𝟏𝟒 𝒙+𝒚− 𝒛=𝟏𝟒

Solve the system.

𝟐 𝒙−𝒚+𝒛=𝟒𝒙+𝟑 𝒚− 𝒛=𝟏𝟏

𝟐 𝒙−𝒚+𝒛=𝟒𝟒 𝒙+𝒚 −𝒛=𝟏𝟒

𝟑 𝒙+𝟐𝒚=𝟏𝟓 𝟔 𝒙=𝟏𝟖𝒙=𝟑

𝟑 (𝟑 )+𝟐 𝒚=𝟏𝟓𝟗+𝟐 𝒚=𝟏𝟓𝟐 𝒚=𝟔𝒚=𝟑

𝟑 𝒙+𝟐𝒚=𝟏𝟓

𝟐 𝒙−𝒚+𝒛=𝟒𝟐(𝟑)−(𝟑)+ 𝒛=𝟒

𝟑+𝒛=𝟒𝒛=𝟏(𝟑 ,𝟑 ,𝟏)

{ 𝒙+𝒚+𝟐 𝒛=𝟑𝟐𝒙+𝒚+𝟑 𝒛=𝟕−𝒙−𝟐 𝒚+𝒛=𝟏𝟎

Solve the system.

𝒙+𝒚+𝟐 𝒛=𝟑 𝟐 𝒙+𝒚+𝟑 𝒛=𝟕−𝒙 −𝟐 𝒚+𝒛=𝟏𝟎−𝒚+𝟑𝒛=𝟏𝟑

−𝟑 𝒚+𝟓 𝒛=𝟐𝟕

−𝟒 𝒛=−𝟏𝟐

−𝒚+𝟑(𝟑)=𝟏𝟑−𝒚+𝟗=𝟏𝟑

𝒙+𝒚+𝟐 𝒛=𝟑𝒙+(−𝟒)+𝟐(𝟑)=𝟑

𝒙−𝟐=𝟑𝒙=𝟏

𝟐 𝒙+𝒚+𝟑 𝒛=𝟕−𝒙 −𝟐 𝒚+𝒛=𝟏𝟎

−𝒚+𝟑𝒛=𝟏𝟑−𝟑 𝒚+𝟓 𝒛=𝟐𝟕

𝒛=𝟑

(𝟏 ,−𝟒 ,𝟑)

−𝟐 𝒙−𝟒 𝒚+𝟐 𝒛=𝟐𝟎

𝟑 𝒚 −𝟗 𝒛=−𝟑𝟗−𝟑 𝒚+𝟓 𝒛=𝟐𝟕

−𝒚=𝟒𝒚=−𝟒

𝑻 +𝑷 +𝑹=𝟐𝟎𝟎𝟏𝟐𝑻 +𝟐𝟒 𝑷+𝟑𝟔𝑹=𝟔𝟎𝟎𝟎𝑹=𝟐𝑷

𝑻 +𝑷 +(𝟐𝑷 )=𝟐𝟎𝟎𝟏𝟐𝑻 +𝟐𝟒 𝑷+𝟑𝟔(𝟐𝑷 )=𝟔𝟎𝟎𝟎

𝑻 +𝟑𝑷=𝟐𝟎𝟎𝟏𝟐𝑻 +𝟗𝟔𝑷=𝟔𝟎𝟎𝟎

−𝟏𝟐𝑻 −𝟑𝟔𝑷=−𝟐𝟒𝟎𝟎𝟏𝟐𝑻 +𝟗𝟔𝑷=𝟔𝟎𝟎𝟎

𝟔𝟎𝑷=𝟑𝟔𝟎𝟎𝑷=𝟔𝟎

𝑹=𝟐(𝟔𝟎)¿𝟏𝟐𝟎𝑻 +(𝟔𝟎)+(𝟏𝟐𝟎)=𝟐𝟎𝟎

𝑻=𝟐𝟎 Pg. 171 #9-11, 30

You manage a clothing store and budget $6000 to restock 200 shirts.You can buy T-shirts for $12 each, polo shirts $24 each and rugby shirts for $36 each. You want to have twice as many rugby shirts as polo shirts. How many of each type shirt should you buy?

Systems with Three Variables

Objective:

I can use matrices to solve systems.

Use the rules below to change figure 1 into figure 2?

Matrix:• A rectangular array of numbers.• Dimensions: rows × columns

Matrix Name

[ 𝐴 ]=[2 5 113 −2 5 ]

Systems to matrices

matrix [1 4 52 5 4]

[1 4 50 −3 −6 ] [1 4 5

0 1 2] [1 0 −30 1 2 ]

{ 𝑥+4 𝑦=52𝑥+5 𝑦=4

-2 × Row 1 + Row 2Put in Row 2

Divide Row 2 by -3Put in Row 2

-4 × Row 2 + Row 1Put in Row 1

(−𝟑 ,𝟐)

Reduced Row Echelon form (rref)

222

111

CyBxA

CyBxA

1A

b

a

10

011B

2A 2B1C

2Cmatrix

Systems to matrices

222

111

CyBxA

CyBxA

1A

b

a

10

011B

2A 2B1C

2Cmatrix

Reduced Row-Echelon Form

[2nd], [x-1], [►], [alpha], [apps]

or

[2nd], [x-1], [►], [▼] to rref( , [enter]

[2nd], [ x-1], [enter], [enter]

[2nd],[x-1],[►] ,[►], [enter]

enter rows [enter]

enter columns [enter]

enter matrix;

[2nd], [mode]

rrefmatrix

rref

[2 1 55 3 13 ]{ 2𝑥+𝑦=5

5 𝑥+3 𝑦=13(𝟐 ,𝟏)

[1 0 20 1 1]

Solve each system using matrices

183

1152

yx

yx

183

1152

110

301

Solution:( , )3 1

12416

224

0

cba

cba

cba

121416

2124

0111Solution:

( , , )1 1

0100

1010

1001

0

Solve each system using matrices

53

724

yx

yx

513

724

5.010

5.101

Solution:( -1.5, -0.5)

12

82

23

zy

zx

zyx

1120

8201

2131Solution:

( -2 , 3, 5)

5100

3010

2001Pg. 179#15-20,24-27,

32,33,36,37

53

724

yx

yx