Sec. 7.3cSec. 7.3c

Let A be the coefficient matrix of a system of n linearequations in n variables given by AX = B, where X is then x 1 matrix of variables and B is the n x 1 matrix ofnumbers of the right-hand side of the equations. If Aexists, then the system of equations has the unique solution

–1

X = A B–1

Write the system of equations as a matrix equation AX = B,with A as the coefficient matrix of the system.

3 9x y z 2 4 1x z

8 5x y z 1 3 1 9

2 0 4 1

8 1 1 5

x

y

z

AX = B:

Write the matrix equation as a system of equations

2 3 2x y z w 2 8 3z w

9 5 9x z w

1 2 3 1 2

0 0 2 8 3

9 0 1 5 9

1 1 6 3 2

x

y

z

w

6 3 2x y z w

Solve the given system using inverse matrices

3 2 0x y 5x y

3 2

1 1A

0

5B

xX

y

3 2x yA X B

x y

To solve for X, apply theinverse of A to both sidesof the matrix equation:

1X A B10

15

Solution:

(x, y) = (10, 15)

Solve the given system using inverse matrices

3 3 6 20x y z 3 10 40x y z

3 3 6

1 3 10

1 3 5

A

20

40

30

B

x

X y

z

3 5 30x y z

Find1X A B

Solution:(x, y, z) = (18, 118/3, 14)

Solve the given system using inverse matrices

4 2 0x y z 2 6x y z

1 4 2

2 1 1

3 3 5

A

0

6

13

B

x

X y

z

3 3 5 13x y z

Find1X A B

Solution:(x, y, z) = (3, –1/2, 1/2)

Solve the given system using inverse matrices

2 2 8x y z 3 2 10x y z w

2 1 2 0

3 2 1 1

2 1 0 3

4 3 2 5

A

8

10

1

39

B

x

yX

z

w

2 3 1x w y

Find1X A B

Solution:(x, y, z, w) =(4, –2, 1, –3)

4 3 2 5 39x y z w

Use a method of your choice to solve the given system.

6x y z 2 2x y z 1 1 1 6

1 1 2 2

Augmented Matrix:

Solution:(x, y, z) = (2 – 1.5z, –4 – 0.5z, z)

1 0 1.5 2

0 1 0.5 4

RREF:

2f x ax bx c

Fitting a parabola to three points. Determine a, b, and c sothat the points (–1, 5), (2, –1), and (3, 13) are on the graph of

How about a diagram to start???

We need f(–1) = 5, f(2) = –1, and f(3) = 13:

3 9 3 13f a b c 2 4 2 1f a b c

1 5f a b c Now, simply solve

this system!!!

24 6 5f x x x (a, b, c) = (4, –6, –5)

Double-check with a graph?

Mixing Solutions. Aileen’s Drugstore needs to prepare a 60-Lmixture that is 40% acid using three concentrations of acid. Thefirst concentration is 15% acid, the second is 35% acid, and thethird is 55% acid. Because of the amounts of acid solution onhand, they need to use twice as much of the 35% solution asthe 55% solution. How much of each solution should they use?

x = liters of 15% solutiony = liters of 35% solutionz = liters of 55% solution

0.15 0.35 0.55 0.40 60x y z 2 0y z

60x y z

Solve the system!!!Need 3.75 L of 15% acid, 37.5 L of 35% acid, and

18.75 L of 55% acid to make 60 L of 40% acid solution.

Manufacturing. Stewart’s metals has three silver alloys on hand.One is 22% silver, another is 30% silver, and the third is 42%silver. How many grams of each alloy is required to produce 80grams of a new alloy that is 34% silver if the amount of 30% alloyused is twice the amount of 22% alloy used?

x = amount of 22% alloyy = amount of 30% alloyz = amount of 42% alloy

2 0x y 0.22 0.30 0.42 27.2x y z

80x y z

Solve the system!!!

Need approximately 14.545g of the 22% alloy,29.091g of the 30% alloy, and 36.364g of the 42% alloy

to make 80g of the 34% alloy.

Vacation Money. Heather has saved $177 to take with her onthe family vacation. She has 51 bills consisting of $1, $5, and$10 bills. If the number of $5 bills is three times the number of$10 bills, find how many of each bill she has.

x = number of $1 billsy = number of $5 billsz = number of $10 bills 3 0y z

5 10 177x y z 51x y z

Solve the system!!!

Heather has 27 one-dollar bills, 18 fives, and 6 tens.