Using Matrices to Solve System of Equations?

MANUFACTURING Fancy French Perfume Company recently had its secret formula divulged. It turned out that it was using, as the three ingredients, rose oil, oil of fermented prunes and alcohol. Moreover, each 22-ounce econosize bottle contained four more ounces of alcohol than oil of fermented prunes and the amount of alcohol was equal to the combined volume of the other two ingredients. How much of each ingredient did it use in an econosize bottle?
Can anybody help explain how to solve this problem using Elementary Row Operation Matrices???

Let r = rose oil, p = fermented prunes (who thought that one up!), and a = alchohol:
“Each 22 oz bottle” means r + p + a = 22
“four more oz of alchohol than prunes.” translates to
a = 4 + p or
-p + a = 4
“alchohol was equal to the combined…” can be written as
a = p + r or
-r + -p + a = 0

1r + 1p + 1a =22
0r + -1p + 1a = 4
-1r + -1p +1a = 0

can be written as a matrix multiplication
1 1 1 r 22
0 -1 1 p = 4
-1 -1 1 a 0

or for row operations, just

1 1 1 22
0 -1 1 4
-1 -1 1 0

Multiply row 3 by -1 and add row 2 to it. Leave result in row 3
1 1 1 22
0 -1 1 4
1 0 0 4

Add row 2 to row 1, then subtract row 3 from row one. Leave the result in row 1.

0 0 2 22
0 -1 1 4
1 0 0 4

Divide row 1 by 2

0 0 1 11
0 -1 1 4
1 0 0 4

Multipy row 2 by -1 then add row 1 to it. Leave in row 2:

0 0 1 11
0 1 0 7
1 0 0 4

Which represents:

0r + 0p + 1a = 11, or a = 11
0r + 1p + 0a = 7 or p = 7
1r + 0p + 0a = 4 or r = 4

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