College

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]
\begin{array}{ccc|c}
1 & 1 & 1 & 27 \\
2 & 3 & 5 & 98 \\
1 & 0 & -1 & -5
\end{array}
[/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 solve this problem, we need to determine how many small, medium, and large yogurts were sold. We're given the following information:

1. A small yogurt costs \[tex]$2, a medium costs \$[/tex]3, and a large costs \[tex]$5.
2. In one hour, the shop sold 27 yogurts for \$[/tex]98.
3. There were five more large yogurts sold than small yogurts.

Let's define the variables:
- Let [tex]\( x \)[/tex] be the number of small yogurts sold.
- Let [tex]\( y \)[/tex] be the number of medium yogurts sold.
- Let [tex]\( z \)[/tex] be the number of large yogurts sold.

From the information provided, we can set up the following system of equations:

1. The total number of yogurts sold:
[tex]\[
x + y + z = 27
\][/tex]

2. The total cost of the yogurts sold:
[tex]\[
2x + 3y + 5z = 98
\][/tex]

3. There were five more large yogurts sold than small yogurts:
[tex]\[
z = x + 5
\][/tex]

Now, let's solve this system step-by-step:

Step 1: Substitute the expression for [tex]\( z \)[/tex] from the third equation into the other equations.

Using [tex]\( z = x + 5 \)[/tex] in the first equation:
[tex]\[
x + y + (x + 5) = 27
\][/tex]
Simplify:
[tex]\[
2x + y + 5 = 27
\][/tex]
Subtract 5 from both sides:
[tex]\[
2x + y = 22 \quad \text{(Equation 4)}
\][/tex]

Step 2: Substitute [tex]\( z = x + 5 \)[/tex] in the second equation:
[tex]\[
2x + 3y + 5(x + 5) = 98
\][/tex]
Distribute and simplify:
[tex]\[
2x + 3y + 5x + 25 = 98
\][/tex]
[tex]\[
7x + 3y = 73 \quad \text{(Equation 5)}
\][/tex]

Step 3: Solve the system of equations formed by Equation 4 and Equation 5:
- Equation 4: [tex]\(2x + y = 22\)[/tex]
- Equation 5: [tex]\(7x + 3y = 73\)[/tex]

Multiply Equation 4 by 3 to eliminate [tex]\( y \)[/tex]:
[tex]\[
6x + 3y = 66
\][/tex]

Subtract this from Equation 5:
[tex]\[
(7x + 3y) - (6x + 3y) = 73 - 66
\][/tex]
[tex]\[
x = 7
\][/tex]

Step 4: Substitute [tex]\( x = 7 \)[/tex] back into Equation 4 to find [tex]\( y \)[/tex]:
[tex]\[
2(7) + y = 22
\][/tex]
[tex]\[
14 + y = 22
\][/tex]
[tex]\[
y = 8
\][/tex]

Step 5: Use [tex]\( x = 7 \)[/tex] in the relation for [tex]\( z \)[/tex]:
[tex]\[
z = x + 5 = 7 + 5 = 12
\][/tex]

So, the store sold 7 small yogurts, 8 medium yogurts, and 12 large yogurts. The correct answer is:

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