can't figure out this matrix problem

Question

A souvenir shop gave away small gifts valued at $5 for sales of $25-$74.99, medium gifts valued at $8 for sales $75 - $149.99, and large gifts valued at $12.50 for sales about $150. The store gave away 102 gifts worth a total of $654 and six times as many small gifts as large gifts. How many of each gift were sold?

Answer 1

Let's write a system of equations for this problem. Let x = number of small gifts, y = number of medium gifts, and z = number of large gifts.

 

x + y + z = 102        (total number of gifts is 102)

5x + 8y + 12.5z = 654        (total value of gifts is $654)

x = 6z        (six times as many small gifts as large gifts)

 

 

 

It seems like you want to set up a matrix. So, rearrange that last equation:

 

x - 6z = 0

x + 0y - 6z = 0

 

 

 

Then, set up a matrix using the equations' coefficients:

 

[1  1    1    |  102]     R1

[5  8  12.5  |  654]     R2

[1  0   -6    |   0   ]     R3

 

 

 

And now use row operations to make the left-hand side into the identity matrix. You can multiply/divide any row by a constant or add/subtract a multiple of a row to/from another.

 

 

 

First, subtract 5(R1) from R2:

 

[1  1    1   |  102]

[0  3  7.5  |  144]

[1  0   -6   |   0  ]

 

Subtract R1 from R3:

 

[1  1   1   |   102]

[0  3  7.5  |  144]

[0  -1  -7  |  -102]

 

Add R3 to R1:

 

[1  0   -6   |   0 ]

[0  3  7.5  | 144]
[0 -1   -7  | -102]

 

Add 1/3(R2) to R3:

 

[1  0   -6   |   0  ]
[0  3   7.5  |  144]

[0  0  -4.5  |  -54]

 

Divide R2 by 3 and R3 by -4.5:

 

[1  0   -6   |  0 ]

[0  1  2.5  |  48]

[0  0    1   |  12]

 

Add 6(R3) to R1:

 

[1  0    0   |  72]

[0  1  2.5  |  48]
[0  0    1   |  12]

 

Subtract 2.5(R3) from R2:

 

[1  0  0  |  72]
[0  1  0  |  18]
[0  0  1  |  12]

 

 

 

Now that the left-hand side is the identity matrix, read your solutions from the right-hand side:

 

x = 72

y = 18

z = 12

 

Plug those values back into the initial equations to check your work.