Word Equations, System as a Matrix

Question

Suppose a burger, a shake and an order of fries cost $9.00. Two shakes and three orders of fries costs $12.00. A burger and two orders of fries costs $8.00.

I was told to use a system of equations and express the system as a matrix. And to use elementary row operations to get this matrix into reduced low echelon form.

What is the cost of the burger?

What is the cost of a shake?

What is the cost of an order of fries?

Answer 1

The augmented matrix is:

1 1 1 | 9

0 2 3 | 12

1 0 2 | 8

We will want to convert the base 3x3 matrix to the identity matrix (1 along the diagonals and zeros everywhere else) using row operations (add or subtract rows, multiply elements of one row by any number and add to another row, multiply entire row by a number).

For example, we want to first change 1 to 0 in row 3 column 1. To do that we would subtract row 1 from row 3 and place the result in row 3:

1 1 1 | 9

0 2 3 | 12

0 -1 1 | -1

we then can change 2 to 1 in row 2 by adding row 3 to row 2 and place the result in row 2.

1 1 1 | 9

0 1 4 | 11

0 -1 1 | -1

we then can change -1 to 0 in row 3 by adding row 2 to row 3 and place the result in row 3.

1 1 1 | 9

0 1 4 | 11

0 0 5 | 10

we can then divide the last row by 2 to get one in the 3rd row and 3rd column

1 1 1 | 9

0 1 4 | 11

0 0 1 | 2

We can continue this process until we have the 3x3 identity matrix and the solution in the 4th column.

Answer 2

b + s + f = 9.00

2s + 3f = 12.00

b + 2f = 8.00

Can you solve for b, s, and f and answer?