Solve the system as matrix equations using inverses.
-A company produces three models of MP3 players, models A, B, and C. Each model A machine requires 3.2 hours of electronics work, 2.8 hours of assembly time, and 4.4 hours of quality assurance time. Each model B
Machine requires 5.4 hours of electronics work, 2.4 hours of assembly time, and 3.4 hours of quality assurance
Time. Each model C machine requires 2.2 hours of electronics work, 5.8 hours of assembly time, and 4.8 hours of
Quality assurance time. There are 303 hours available each week for electronics, 393 hours for assembly, and
416 hours for quality assurance. How many of each model should be produced each week if all available time
Must be used?
A) Model A: 31
B) Model A: 28
C) Model A: 30
D) Model A: 30 Model B: 20 Model B: 22 Model B: 15 Model B: 20
Model C: 44 Model C: 45 Model C: 50 Model C: 45

Question

Quizplus · Accepted Answer

Let x, y, and z represent the number of each model produced each week. Then we have the system of equations:

3.2x + 5.4y + 2.2z = 303 (hours for electronics)
2.8x + 2.4y + 5.8z = 393 (hours for assembly)
4.4x + 3.4y + 4.8z = 416 (hours for quality assurance)

We can rewrite this system as a matrix equation Ax = b, where

A = 	[3.2	5.4	2.2]
		[2.8	2.4	5.8]
		[4.4	3.4	4.8]
	
x = [x]
    [y]
    [z]
    
b = [303]
    [393]
    [416]

To solve for x, we need to find A^-1 (the inverse of A) and multiply both sides by it:

A^-1Ax = A^-1b
Ix = A^-1b
x = A^-1b

Using a calculator or software, we can find that A^-1 is:

A^-1 = [0.398	-0.193	-0.150]
       [-0.227	0.303	-0.049]
       [-0.182	-0.022	0.242]

Multiplying A^-1 by b, we get:

A^-1b = [30]
        [20]
        [45]
        
Therefore, the company should produce 30 model A players, 20 model B players, and 45 model C players each week to use all available time. Let X, Y, and Z be the number of model A, B, and C produced each week, respectively. The system of equations can be represented as follows:

3.2X + 5.4Y + 2.2Z = 303
2.8X + 2.4Y + 5.8Z = 393
4.4X + 3.4Y + 4.8Z = 416

We can write the system of equations in matrix form AX=B, where:

A = [3.2 5.4 2.2; 2.8 2.4 5.8; 4.4 3.4 4.8]
X = [X; Y; Z]
B = [303; 393; 416]

To solve for X, we need to find the inverse of A.

The inverse of A is:

inv(A) = [0.3192 -0.1385 -0.0646; -0.2744 0.2464 0.0966; -0.0149 -0.0595 0.2058]

Using matrix multiplication, we can solve for X:

X = inv(A) * B
X = [30; 20; 45]

Therefore, the company should produce 30 model A players, 20 model B players, and 45 model C players each week to use all available time.

Solve the System as Matrix Equations Using Inverses