High School

A store sells yogurt in small, medium, and large sizes. A small yogurt costs [tex]\$2[/tex], a medium costs [tex]\$3[/tex], and a large costs [tex]\$5[/tex]. In one hour, the shop sold 27 yogurts for [tex]\$98[/tex]. There were five more large yogurts than small yogurts. The matrix below represents the system of equations for this situation:

[tex]
\left[\begin{array}{ccc|c}
1 & 1 & 1 & 27 \\
2 & 3 & 5 & 98 \\
1 & 0 & -1 & -5
\end{array}\right]
[/tex]

How many of each size yogurt did the store sell during that hour?

A. 7 small yogurts, 8 medium yogurts, 12 large yogurts
B. 12 small yogurts, 8 medium yogurts, 7 large yogurts
C. 27 small yogurts, 98 medium yogurts, 5 large yogurts
D. 98 small yogurts, 27 medium yogurts, 5 large yogurts

Answer :

To determine how many of each size yogurt the store sold during the hour, we need to solve the system of equations represented by the following matrix:
[tex]\[
\left[\begin{array}{ccc|c}
1 & 1 & 1 & 27 \\
2 & 3 & 5 & 98 \\
1 & 0 & -1 & -5
\end{array}\right]
\][/tex]

This matrix represents the following system of equations:
1. [tex]\( x + y + z = 27 \)[/tex]
2. [tex]\( 2x + 3y + 5z = 98 \)[/tex]
3. [tex]\( x - z = -5 \)[/tex]

Here, [tex]\( x \)[/tex] represents the number of small yogurts, [tex]\( y \)[/tex] represents the number of medium yogurts, and [tex]\( z \)[/tex] represents the number of large yogurts. Let's solve this step-by-step:

Step 1: Solve the third equation for [tex]\( x \)[/tex]:
[tex]\[ x - z = -5 \][/tex]
[tex]\[ x = z - 5 \][/tex]

Step 2: Substitute [tex]\( x \)[/tex] in the other equations:
Substitute [tex]\( x = z - 5 \)[/tex] into the first and the second equations.

For the first equation [tex]\( x + y + z = 27 \)[/tex]:
[tex]\[ (z - 5) + y + z = 27 \][/tex]
[tex]\[ 2z + y - 5 = 27 \][/tex]
[tex]\[ 2z + y = 32 \][/tex]
[tex]\[ y = 32 - 2z \][/tex]

For the second equation [tex]\( 2x + 3y + 5z = 98 \)[/tex]:
[tex]\[ 2(z - 5) + 3y + 5z = 98 \][/tex]
[tex]\[ 2z - 10 + 3y + 5z = 98 \][/tex]
[tex]\[ 7z + 3y - 10 = 98 \][/tex]
[tex]\[ 7z + 3y = 108 \][/tex]
[tex]\[ 3y = 108 - 7z \][/tex]
[tex]\[ y = \frac{108 - 7z}{3} \][/tex]

Step 3: Substitute [tex]\( y = 32 - 2z \)[/tex] into [tex]\( y = \frac{108 - 7z}{3} \)[/tex] to find [tex]\( z \)[/tex]:
[tex]\[ 32 - 2z = \frac{108 - 7z}{3} \][/tex]

Clear the fraction by multiplying through by 3:
[tex]\[ 3(32 - 2z) = 108 - 7z \][/tex]
[tex]\[ 96 - 6z = 108 - 7z \][/tex]
[tex]\[ 96 + z = 108 \][/tex]
[tex]\[ z = 108 - 96 \][/tex]
[tex]\[ z = 12 \][/tex]

Step 4: Find [tex]\( y \)[/tex] using [tex]\( y = 32 - 2z \)[/tex]:
[tex]\[ y = 32 - 2(12) \][/tex]
[tex]\[ y = 32 - 24 \][/tex]
[tex]\[ y = 8 \][/tex]

Step 5: Find [tex]\( x \)[/tex] using [tex]\( x = z - 5 \)[/tex]:
[tex]\[ x = 12 - 5 \][/tex]
[tex]\[ x = 7 \][/tex]

Thus, the store sold:
- 7 small yogurts
- 8 medium yogurts
- 12 large yogurts

So the correct answer is:
7 small yogurts, 8 medium yogurts, 12 large yogurts.