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The problem you're working on involves a proportional relationship between the amount of cleaning solution \( x \) (in quarts) and the area of carpet \( y \) (in square yards) that Mike cleans with this solution.

According to the information provided:
- Mike uses **2 quarts** of cleaning solution to clean **38 square yards** of carpet.

Since the relationship is proportional, the equation that represents the relationship between \( x \) and \( y \) can be written in the form:
\[
y = kx
\]
where \( k \) is the constant of proportionality. To find \( k \), we use the given values \( x = 2 \) and \( y = 38 \):

\[
38 = k \times 2
\]
Solving for \( k \):
\[
k = \frac{38}{2} = 19
\]

Therefore, the equation for the relationship between \( x \) (quarts of solution) and \( y \) (square yards of carpet cleaned) is:
\[
y = 19x
\]

This equation means that for each quart of cleaning solution, Mike can clean 19 square yards of carpet.

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Do you want further details or have any questions? Here are 5 related questions to expand on this topic:

1. How would the equation change if Mike used 3 quarts of cleaning solution for 60 square yards of carpet?
2. What is the area cleaned if Mike uses 5 quarts of cleaning solution?
3. Can you express \( x \) in terms of \( y \) in this relationship?
4. How would the equation change if the relationship was inversely proportional?
5. What would happen to the area cleaned if the amount of cleaning solution used is doubled?

**Tip:** Always check if the relationship in word problems is proportional by comparing ratios for different quantities!

Mike owns and operates Stain-B-Gone Carpet Cleaners. He uses a special non-toxic cleaning solution to clean carpets. There is a proportional relationship between the amount of cleaning solution (in quarts) Mike uses, x, and the area of carpet (in square yards) he cleans with the solution, y. Mike uses 2 quarts of cleaning solution to clean 38 square yards of carpet. Write the equation for the relationship between x and y.

This problem involves finding a proportional relationship between the amount of cleaning solution used (in quarts) and the area cleaned (in square yards). Mike uses 2 quarts of cleaning solution to clean 38 square yards of carpet. The equation y = 19x shows that for each quart of solution, Mike can clean 19 square yards.

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