1. Define variables: Let \( x \) = number of cars washed, \( y \) = number of windows cleaned.
2. Write inequalities:
- Total earnings: \( 5x + 3y \geq 500 \)
- Time constraint: \( x + y \leq 100 \)
- Non-negativity: \( x \geq 0 \), \( y \geq 0 \)
3. Graph each inequality on the coordinate plane.
4. Shade the feasible region where all inequalities overlap.
5. Identify corner points of the feasible region: (0, 100), (100, 0), and intersection of \( 5x + 3y = 500 \) and \( x + y = 100 \).
6. Solve for intersection: Substitute \( y = 100 - x \) into \( 5x + 3(100 - x) = 500 \) → \( 5x + 300 - 3x = 500 \) → \( 2x = 200 \) → \( x = 100 \), so \( y = 0 \). Thus, corner points are (0, 100) and (100, 0).
7. Verify solutions: At (0, 100): $5(0) + 3(100) = 300 < 500$ — not feasible. At (100, 0): $5(100) + 3(0) = 500 \geq 500$ — feasible.
8. Conclusion: The only feasible solution with maximum work is washing 100 cars and cleaning 0 windows. Other combinations require more than 100 total tasks or don’t meet the $500 goal.
Parent Tip: Review the logic above to help your child master the concept of systems inequalities worksheet.
