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Monica's school band held a car wash to raise money for a trip to a parade in New York City. After washing 125 cars, they made \[tex]$775 from a combination of \$[/tex]5.00 quick washes and \$8.00 premium washes.

This system of equations models the situation:
[tex]\[
\begin{aligned}
x + y &= 125 \\
5x + 8y &= 775
\end{aligned}
\][/tex]

Solve the system to answer the questions:

1. How many premium car washes were ordered?
[tex]\(\quad \_\_\_\_\_\)[/tex] premium car washes

2. How many quick car washes were ordered?
[tex]\(\quad \_\_\_\_\_\)[/tex] quick car washes



Answer :

GinnyAnswer
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

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