Question

Write the system of linear equations in \( x \) and \( y \) formed from the given situation.

Hint:

To form equations from a word problem: 1. Use the condition that the total remains constant (e.g., total donation). 2. Write separate equations for the given scenarios and simplify them.

Answer 1

Step 1: The total amount donated is consistent across both scenarios. - Scenario 1: With \( x - 8 \) children, each receives \( y + 10 \). The total donation is represented by: \[(x - 8)(y + 10) = xy.\]Upon simplification: \[xy - 8y + 10x - 80 = xy \quad \Rightarrow \quad -8y + 10x = 80 \quad \Rightarrow \quad 10x - 8y = 80. \quad \cdots (1)\]- Scenario 2: With \( x + 16 \) children, each receives \( y - 10 \). The total donation is represented by: \[(x + 16)(y - 10) = xy.\]Upon simplification: \[xy + 16y - 10x - 160 = xy \quad \Rightarrow \quad 16y - 10x = 160. \quad \cdots (2)\]Step 2: The resulting system of linear equations is: \[10x - 8y = 80, \quad -10x + 16y = 160.\]