To solve this system of equations, we need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] where:
[tex]\[ x + y = 125 \][/tex]
[tex]\[ 5x + 8y = 775 \][/tex]
Here, [tex]\( x \)[/tex] represents the number of quick car washes, and [tex]\( y \)[/tex] represents the number of premium car washes.
### Step-by-Step Solution:
1. Express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
x + y = 125 \quad \Rightarrow \quad y = 125 - x
\][/tex]
2. Substitute [tex]\( y \)[/tex] into the second equation:
[tex]\[
5x + 8(125 - x) = 775
\][/tex]
3. Distribute the 8 inside the parenthesis:
[tex]\[
5x + 1000 - 8x = 775
\][/tex]
4. Combine like terms:
[tex]\[
5x - 8x + 1000 = 775 \quad \Rightarrow \quad -3x + 1000 = 775
\][/tex]
5. Isolate [tex]\( x \)[/tex]:
[tex]\[
-3x + 1000 = 775 \quad \Rightarrow \quad -3x = 775 - 1000
\][/tex]
[tex]\[
-3x = -225
\][/tex]
[tex]\[
x = \frac{-225}{-3} \quad \Rightarrow \quad x = 75
\][/tex]
So, the number of quick car washes ([tex]\( x \)[/tex]) is 75.
6. Use [tex]\( x \)[/tex] to find [tex]\( y \)[/tex]:
[tex]\[
y = 125 - x \quad \Rightarrow \quad y = 125 - 75 \quad \Rightarrow \quad y = 50
\][/tex]
Therefore, the number of premium car washes ([tex]\( y \)[/tex]) is 50.
### Final Answer:
- Premium car washes: 50
- Quick car washes: 75
