To solve the given problem, we need to analyze the system of equations provided:
1. [tex]\( x + y = 24 \)[/tex]
2. [tex]\( 3.50x + 5.00y = 97.50 \)[/tex]
Here, [tex]\( x \)[/tex] represents the number of 10 oz. boxes sold, and [tex]\( y \)[/tex] represents the number of 16 oz. boxes sold.
### Step-by-Step Solution:
1. Equation (1): [tex]\( x + y = 24 \)[/tex]
From this equation, we can express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y = 24 - x
\][/tex]
2. Equation (2): [tex]\( 3.50x + 5.00y = 97.50 \)[/tex]
Substitute the expression for [tex]\( y \)[/tex] from Equation (1) into Equation (2):
[tex]\[
3.50x + 5.00(24 - x) = 97.50
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
3.50x + 120 - 5.00x = 97.50
\][/tex]
Combine like terms:
[tex]\[
3.50x - 5.00x + 120 = 97.50
\][/tex]
[tex]\[
-1.50x + 120 = 97.50
\][/tex]
Isolate [tex]\( x \)[/tex]:
[tex]\[
-1.50x = 97.50 - 120
\][/tex]
[tex]\[
-1.50x = -22.50
\][/tex]
Divide by -1.50:
[tex]\[
x = \frac{-22.50}{-1.50} = 15
\][/tex]
Thus, [tex]\( x = 15 \)[/tex].
4. Solve for [tex]\( y \)[/tex]:
Using the first equation [tex]\( y = 24 - x \)[/tex]:
[tex]\[
y = 24 - 15
\][/tex]
[tex]\[
y = 9
\][/tex]
Therefore, Jillian sold:
- [tex]\( x = 15 \)[/tex] 10 oz. boxes
- [tex]\( y = 9 \)[/tex] 16 oz. boxes
Hence, the number of 10 oz. boxes sold is [tex]\( 15 \)[/tex]. Therefore, the correct answer is:
15
