3x + 2y = 183x - y = 9What is the solution \((\mathrm{x},\mathrm{y})\) to the given system of equations?

1. TRANSLATE the problem information

• Given system:
  • \(3\mathrm{x} + 2\mathrm{y} = 18\)
  • \(3\mathrm{x} - \mathrm{y} = 9\)
• Need to find: The solution \((\mathrm{x},\mathrm{y})\) that satisfies both equations

2. INFER the most efficient solution method

• Notice both equations have the same coefficient for x \((3\mathrm{x})\)
• This makes elimination the ideal approach - we can eliminate x by subtracting one equation from the other
• Strategy: Subtract the second equation from the first

3. SIMPLIFY through elimination

• Set up: \((3\mathrm{x} + 2\mathrm{y}) - (3\mathrm{x} - \mathrm{y}) = 18 - 9\)
• Distribute the negative carefully: \(3\mathrm{x} + 2\mathrm{y} - 3\mathrm{x} + \mathrm{y} = 9\)
• Combine like terms: \(3\mathrm{y} = 9\)
• Solve: \(\mathrm{y} = 3\)

4. SIMPLIFY by substitution to find x

• Substitute \(\mathrm{y} = 3\) into either original equation (using the second equation):
• \(3\mathrm{x} - 3 = 9\)
• Add 3: \(3\mathrm{x} = 12\)
• Divide by 3: \(\mathrm{x} = 4\)

5. Verify the solution

• Check in both equations:
  • First equation: \(3(4) + 2(3) = 12 + 6 = 18\) ✓
  • Second equation: \(3(4) - 3 = 12 - 3 = 9\) ✓

Answer: \((4,3)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when distributing the negative in the elimination step.

When subtracting \((3\mathrm{x} - \mathrm{y})\), they might write: \(3\mathrm{x} + 2\mathrm{y} - 3\mathrm{x} - \mathrm{y} = 9\) instead of \(3\mathrm{x} + 2\mathrm{y} - 3\mathrm{x} + \mathrm{y} = 9\). This leads to \(\mathrm{y} = 9\) instead of \(3\mathrm{y} = 9\), giving \(\mathrm{y} = 9\). Substituting back gives \(\mathrm{x} = -3\), leading to the incorrect solution \((-3,9)\), which doesn't match any answer choice and causes confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students choose substitution instead of elimination and make computational errors.

They might solve the second equation for y: \(\mathrm{y} = 3\mathrm{x} - 9\), then substitute into the first equation. While this approach works, it creates more complex arithmetic \((3\mathrm{x} + 2(3\mathrm{x} - 9) = 18)\), increasing chances for calculation mistakes. Students might incorrectly simplify to get \(\mathrm{x} = 3\), then \(\mathrm{y} = 4\), leading them to select Choice B \((3,4)\).

The Bottom Line:

This problem tests whether students can execute the elimination method cleanly, particularly handling negative signs correctly when subtracting equations. The key insight is recognizing that elimination is more efficient than substitution when coefficients align perfectly.