Question

A store sells yogurt in small, medium, and large sizes. A small yogurt costs $2, a medium costs $3, and a large costs $5. In one hour, the shop sold 27 yogurts for $98. There were five more large yogurts than small yogurts. The matrix below represents the system of equations in this situation
How many of each size yogurt did the store sell during that hour?
7 small yogurts, 8 medium yogurts, 12 large yogurts12 small yogurts, 8 medium yogurts, 7 large yogurts27 small yogurts, 98 medium yogurts, 5 large yogurts98 small yogurts, 27 medium yogurts, 5 large yogurts

Answer 1

Let S = number of small yogurts ($2 each).
Let M = number of medium yogurts ($3 each)
Let L = number of large yogurts ($5 each)

Total yogurts is 27, therefore
S + M + L = 27

Total revenue generated is $98, therefore
2S + 3M +5L = 98

There are five more large yogurts than small yogurts, therefore
L = S + 5, or
-S + L = 5

These three equations may be written as the matrix equation
[  1   1   1  | |S|      |27|
|  2  3  5 | |M| =   |98|
| -1   0  1 | |L|       | 5|

The determinant of the matrix is
D = 3 - (2+5) + 3 = -1.

Solve with Cramer's Rule to obtain
S = -[27*3) - (98-25) - 15]
   = 7
M = -[(98-25) - 27(2+5) + (10+98)]
    = 8
L = -[15 - (10+98) + 27(3)]
   = 12

Answer: 7 small, 8 medium, 12 large yogurts.