Question:1/3a + 1/4b = 1/2b = -2aWhat is the solution (a,b) to the given system of equations?\((-3, 6)\)\((-\frac{1}{4}, \frac{1}{2})\)\((3, -6)\)\((6...

1. TRANSLATE the problem information

• Given system:
  • \(\frac{1}{3}\mathrm{a} + \frac{1}{4}\mathrm{b} = \frac{1}{2}\)
  • \(\mathrm{b} = -2\mathrm{a}\)
• What this tells us: The second equation is already solved for b, making substitution the natural choice

2. INFER the most efficient approach

• Since \(\mathrm{b} = -2\mathrm{a}\) is already isolated, substitute this expression for b into the first equation
• This will give us one equation with only variable a

3. SIMPLIFY by substitution and combining terms

• Substitute \(\mathrm{b} = -2\mathrm{a}\) into the first equation:
  \(\frac{1}{3}\mathrm{a} + \frac{1}{4}(-2\mathrm{a}) = \frac{1}{2}\)
• Simplify the fraction multiplication:
  \(\frac{1}{3}\mathrm{a} - \frac{2}{4}\mathrm{a} = \frac{1}{2}\)
  \(\frac{1}{3}\mathrm{a} - \frac{1}{2}\mathrm{a} = \frac{1}{2}\)

4. SIMPLIFY fraction arithmetic

• Find common denominator for 1/3 and 1/2 (LCD is 6):
  \(\frac{2}{6}\mathrm{a} - \frac{3}{6}\mathrm{a} = \frac{1}{2}\)
• Combine like terms:
  \(-\frac{1}{6}\mathrm{a} = \frac{1}{2}\)

5. SIMPLIFY to solve for a

• Multiply both sides by -6:
  \(\mathrm{a} = \frac{1}{2} \times (-6) = -3\)

6. SIMPLIFY to find b

• Substitute \(\mathrm{a} = -3\) into \(\mathrm{b} = -2\mathrm{a}\):
  \(\mathrm{b} = -2(-3) = 6\)

7. INFER verification is needed

• Check both original equations:
  • First: \(\frac{1}{3}(-3) + \frac{1}{4}(6) = -1 + 1.5 = 0.5 = \frac{1}{2}\) ✓
  • Second: \(6 = -2(-3) = 6\) ✓

Answer: A (-3, 6)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Making arithmetic errors when working with fractions, particularly when finding common denominators or combining \(-\frac{1}{3}\mathrm{a}\) and \(\frac{1}{2}\mathrm{a}\)

Students might incorrectly combine these as \(-\frac{1}{5}\mathrm{a}\) (adding denominators) or make sign errors, getting \(\frac{1}{6}\mathrm{a} = \frac{1}{2}\) instead of \(-\frac{1}{6}\mathrm{a} = \frac{1}{2}\). This leads to \(\mathrm{a} = 3\) instead of \(\mathrm{a} = -3\), and consequently \(\mathrm{b} = -6\).

This may lead them to select Choice C (3, -6)

Second Most Common Error:

Poor SIMPLIFY execution: Making a sign error when multiplying by -6 or when substituting the negative value

Some students correctly get to \(-\frac{1}{6}\mathrm{a} = \frac{1}{2}\) but then multiply incorrectly, getting \(\mathrm{a} = 3\), or they make an error when computing \(\mathrm{b} = -2(-3)\), forgetting the double negative rule.

This leads to confusion and potentially selecting Choice C (3, -6) or other incorrect answers

The Bottom Line:

This problem tests precision with fraction arithmetic and sign management. The algebra isn't conceptually difficult, but the multiple fraction operations create many opportunities for computational errors that lead directly to the wrong answer choices.