To determine the correct system of linear equations representing the situation where Monica's school band raised money through car washes, let's analyze the problem step-by-step:
1. Define Variables:
- Let [tex]\( x \)[/tex] be the number of quick washes.
- Let [tex]\( y \)[/tex] be the number of premium washes.
2. Total Cars Washed:
- According to the problem, the total number of cars washed is 125. This gives us our first equation:
[tex]\[
x + y = 125
\][/tex]
3. Total Money Raised:
- The school band raised a total of [tex]$775 from the car washes.
- The quick wash costs $[/tex]5.00, so the money raised from the quick washes is [tex]\( 5x \)[/tex].
- The premium wash costs $8.00, so the money raised from the premium washes is [tex]\( 8y \)[/tex].
- The total money raised from both types of washes is:
[tex]\[
5x + 8y = 775
\][/tex]
4. Form the System of Equations:
- Combining the equations derived from the total number of cars washed and the total money raised, we have the system of linear equations:
[tex]\[
\begin{cases}
x + y = 125 \\
5x + 8y = 775
\end{cases}
\][/tex]
5. Compare with Given Options:
- Compare our derived system with the given options:
1. [tex]\( 5x + 8y = 775 \)[/tex] and [tex]\( x + y = 125 \)[/tex]
2. [tex]\( 5x - 8y = 125 \)[/tex] and [tex]\( x + y = 775 \)[/tex]
3. [tex]\( 5x + 8y = 775 \)[/tex] and [tex]\( x - y = 125 \)[/tex]
4. [tex]\( 5x - 8y = 125 \)[/tex] and [tex]\( x - y = 775 \)[/tex]
- The system we derived matches the first option:
[tex]\[
\boxed{1}\; \text{(5x + 8y = 775 and x + y = 125)}
\][/tex]
Hence, the correct system of linear equations representing the situation is:
[tex]\[ \boxed{1} \][/tex]
