The Moore family and the Walker family each used their sprinklers last summer. The water output rate for the Moore family's sprinkler was 40 L per hour. The water output rate for the Walker family's sprinkler was 35 L per hour. The families used their sprinklers for a combined total of 55 hours, resulting in a total water output of 2100 L. How long was each sprinkler used?

To determine how long each family's sprinkler was used, let's define variables for the hours each sprinkler was used:
- Let \( x \) be the number of hours the Moore family's sprinkler was used.
- Let \( y \) be the number of hours the Walker family's sprinkler was used.

We have two pieces of information that can be translated into a system of equations:
1. The combined total time the sprinklers were used: 
\[ x + y = 55 \]
2. The combined total water output:
\[ 40x + 35y = 2100 \]

We now have the following system of linear equations:
\[
\begin{cases}
x + y = 55 \\
40x + 35y = 2100
\end{cases}
\]

First, we can solve the first equation for \( y \):
\[ y = 55 - x \]

Next, we substitute \( y \) in the second equation:
\[
40x + 35(55 - x) = 2100
\]

Expand and simplify:
\[
40x + 1925 - 35x = 2100
\]

Combine like terms:
\[
5x + 1925 = 2100
\]

Subtract 1925 from both sides:
\[
5x = 175
\]

Divide both sides by 5:
\[
x = 35
\]

Now, substitute \( x = 35 \) back into the equation \( y = 55 - x \):
\[
y = 55 - 35 = 20
\]

So, the Moore family's sprinkler was used for \( 35 \) hours and the Walker family's sprinkler was used for \( 20 \) hours.

To verify, we can check the total water output:
\[
40x + 35y = 40(35) + 35(20) = 1400 + 700 = 2100 \text{ L}
\]

The calculations are correct. Therefore, the Moore family's sprinkler was used for **35 hours** and the Walker family's sprinkler was used for **20 hours**.

Would you like further details or have any questions?

Here are 8 related questions that expand on this problem:
1. If the Moore family's sprinkler had an output rate of 50 L per hour, how would the hours change?
2. What if the Walker family used their sprinkler for 5 hours longer than the Moore family, how would you set up the equations?
3. How would the solution change if the total combined hours used was 60 instead of 55?
4. What if the total water output was 2500 L instead of 2100 L, how would that affect the solution?
5. Can you solve the system using matrix methods?
6. What if the Moore family's sprinkler broke down after 20 hours, how many hours would the Walker family's sprinkler need to cover the remaining water output?
7. How would the solution change if both families used their sprinklers at different rates each hour?
8. Can you express the time each family used their sprinkler as a percentage of the total time?

**Tip:** When solving systems of equations, substituting one equation into the other can simplify the problem, making it easier to solve for the unknowns.

Learn how to solve the problem of determining the usage time for sprinklers based on their water output rates. This problem involves solving a system of linear equations where each equation represents the total hours and water output for the Moore and Walker families' sprinklers. Suitable for students in Grades 7-9.

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