Matrices word problem?

Question

A group of students decides to sell pizzas to help raise money for their senior class trip. They sold pepperoni for $12, sausage for $10, and cheese for $8. At the end of their sales the class sold a total of 600 pizzas and made $5900. The students sold 175 more cheese pizzas than sausage pizzas. Set up a system of three equations and three unknowns, use an augmented matrix to solve, and show all work to receive full credit.

A. What are the three unknowns?

B. Write a separate equation representing each of the first three sentences.

C. Determine the augmented matrix that represents the three equations.

D. Solve for the matrix. Show all work.


E. How many of each type of pizza were sold?

Answer 1

Let p be # of pepperoni pizzas
Let s be # of sausage pizzas
Let c be # of cheese pizzas

Since sold 600 pizzas total:
p + s + c = 600

Since made $5900 for all pizzas (with prices given in question):
12*p + 10*s + 8*c = 5900

And since 175 more cheese than sausage pizzas:
c - s = 175, or
c - s + 0*p = 175

Thus, the equations are:
12*p + 10*s + 8*c = 5900
1*p + 1*s + 1*c = 600
0*p - 1*s + 1*c = 175

Augmented matrix is:
12 10 08 | 5900
1 1 1 | 600
0 -1 1 | 175

To solve, you can convert to row reduced form (rrf). Basically, subtract a multiple of one row from another row. Keep doing this until you get an upper triangular shape (this will be demonstrated...)

Note: For an algebra class, make sure you use the techniques they teach in the class...

So, start with:

12 10 08 | 5900
1 1 1 | 600
0 -1 1 | 175

Subtract 11*(row 2) from (row 1)

1 -1 -3 | -700
1 1 1 | 600
0 -1 1 | 175

Subtract 1*(row 1) from (row 2)

1 -1 -3 | -700
0 2 4 | 1300
0 -1 1 | 175

Add 0.5*(row 2) to (row 3)

1 -1 -3 | -700
0 2 4 | 1300
0 0 3 | 825

Above is in row-reduced form (can see that the non-zero numbers form an upper triangle). This is called an upper triangular matrix. Anyways...

Now to solve:

Using row 3:
3c = 825
c = 275

Using row 2 (and c=275)
2s + 4c = 1300
s + 2c = 650
s = 650 - 2*(275)
s = 650 - 550
s = 100

Using row 1 (and c=275, s =100)
p - s - 3c = -700
p = -700 + s + 3c
p = -700 + 100 + 825
p = 225

So there were 225 pepperoni, 100 sausage, and 275 cheese pizzas sold...

Again, the above just shows the idea. Use the techniques taught in the class (I think a lot of teachers are particular about the process used to reduce the matrix)
.
.

Answer 2

p + s + c = 600
12p + 10s + 8c = 5900
s - c = 175

You should be able to take it from there.