a = b - 4 5a + 3b = 12 The solution to the given system of equations is \(\mathrm{(a,b)}\)....

1. TRANSLATE the problem information

• Given system:
  • Equation 1: \(\mathrm{a = b - 4}\)
  • Equation 2: \(\mathrm{5a + 3b = 12}\)
• Find: the value of b

2. INFER the best solution strategy

• Since equation 1 already gives us a in terms of b, substitution is the most direct approach
• We can substitute the expression for a into equation 2 to get an equation with only b

3. SIMPLIFY by substituting and solving

• Substitute \(\mathrm{(b - 4)}\) for a in equation 2:
  \(\mathrm{5(b - 4) + 3b = 12}\)
• Apply distributive property:
  \(\mathrm{5b - 20 + 3b = 12}\)
• Combine like terms:
  \(\mathrm{8b - 20 = 12}\)
• Add 20 to both sides:
  \(\mathrm{8b = 32}\)
• Divide both sides by 8:
  \(\mathrm{b = 4}\)

4. Verify the solution

• If \(\mathrm{b = 4}\), then \(\mathrm{a = 4 - 4 = 0}\)
• Check: \(\mathrm{5(0) + 3(4) = 0 + 12 = 12}\) ✓

Answer: b = 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors during the multi-step simplification process, particularly when applying the distributive property or combining like terms.

For example, they might incorrectly distribute as \(\mathrm{5(b - 4) = 5b - 4}\) instead of \(\mathrm{5b - 20}\), or make arithmetic errors when combining \(\mathrm{5b + 3b = 8b}\). These calculation mistakes lead to wrong final values for b.

This leads to confusion and incorrect answer selection.

Second Most Common Error:

Poor TRANSLATE reasoning: Students solve for the wrong variable, finding \(\mathrm{a = 0}\) instead of recognizing the question asks for b.

Even with correct algebraic work, they might stop after finding \(\mathrm{a = 0}\) and select this as their answer, missing that the question specifically asks for the value of b.

This causes them to provide an incorrect response despite using valid solution methods.

The Bottom Line:

This problem tests both strategic thinking (choosing substitution) and careful algebraic execution. Success requires recognizing the most efficient approach and then executing multiple algebraic steps without errors.