Consider the system of equations: 3a + 2b = 17 a + b = 5 If \(\mathrm{(a, b)}\) represents the...

1. INFER the best solution approach

• We have two equations with two unknowns
• The second equation (\(\mathrm{a + b = 5}\)) is simpler, so we can easily isolate one variable
• Substitution method will be most efficient here

2. SIMPLIFY to isolate one variable

• From \(\mathrm{a + b = 5}\), subtract a from both sides:
• \(\mathrm{b = 5 - a}\)

3. SIMPLIFY by substituting into the first equation

• Replace b in the first equation: \(\mathrm{3a + 2(5 - a) = 17}\)
• Distribute the 2: \(\mathrm{3a + 10 - 2a = 17}\)
• Combine like terms: \(\mathrm{a + 10 = 17}\)
• Subtract 10: \(\mathrm{a = 7}\)

4. Verify the solution (always recommended)

• If \(\mathrm{a = 7}\), then \(\mathrm{b = 5 - 7 = -2}\)
• Check: \(\mathrm{3(7) + 2(-2) = 21 - 4 = 17}\) ✓
• Check: \(\mathrm{7 + (-2) = 5}\) ✓

Answer: D) 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign errors during distribution or combining like terms

When distributing \(\mathrm{2(5 - a)}\), students may incorrectly get \(\mathrm{3a + 10 + 2a = 17}\) (forgetting the negative sign), leading to \(\mathrm{5a = 7}\), so \(\mathrm{a = 7/5}\). Since \(\mathrm{7/5}\) isn't among the choices, this leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Attempting elimination method unnecessarily

Some students multiply equations to eliminate variables rather than using the simpler substitution approach. This creates more complex arithmetic like multiplying the second equation by -2 or -3, increasing chances for computational errors that could lead them to select incorrect answer choices.

The Bottom Line:

This problem tests whether students can execute a straightforward substitution cleanly. The algebraic steps are not complex, but each step requires careful attention to signs and arithmetic—one small error cascades through to an incorrect final answer.