(a + 5)/2 = 3b - cThe given equation relates the positive numbers a, b, and c. Which equation correctly...

1. INFER the solution strategy

• We have: \(\frac{\mathrm{a + 5}}{2} = 3\mathrm{b} - \mathrm{c}\)
• Goal: Express a in terms of b and c
• Strategy: Eliminate the fraction first, then isolate a

2. SIMPLIFY by eliminating the fraction

• Multiply both sides by 2:
  \(\frac{\mathrm{a + 5}}{2} \times 2 = (3\mathrm{b} - \mathrm{c}) \times 2\)
• This gives us: \(\mathrm{a + 5} = 2(3\mathrm{b} - \mathrm{c})\)

3. SIMPLIFY using the distributive property

• Distribute the 2: \(\mathrm{a + 5} = 6\mathrm{b} - 2\mathrm{c}\)
• Now we have a linear equation without fractions

4. SIMPLIFY to isolate a

• Subtract 5 from both sides: \(\mathrm{a} = 6\mathrm{b} - 2\mathrm{c} - 5\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make distribution errors when expanding \(2(3\mathrm{b} - \mathrm{c})\)

Many students incorrectly distribute to get \(2(3\mathrm{b} - \mathrm{c}) = 6\mathrm{b} - \mathrm{c}\) (forgetting to multiply the -c by 2), leading to:
\(\mathrm{a + 5} = 6\mathrm{b} - \mathrm{c}\)
\(\mathrm{a} = 6\mathrm{b} - \mathrm{c} - 5\)

This may lead them to select Choice B (\(\mathrm{a} = 6\mathrm{b} - \mathrm{c} - 5\))

Second Most Common Error:

Poor SIMPLIFY execution: Sign errors when moving the constant term

Students correctly get to \(\mathrm{a + 5} = 6\mathrm{b} - 2\mathrm{c}\), but then make a sign error when subtracting 5, thinking:
\(\mathrm{a} = 6\mathrm{b} - 2\mathrm{c} + 5\)

This may lead them to select Choice C (\(\mathrm{a} = 6\mathrm{b} - 2\mathrm{c} + 5\))

The Bottom Line:

This problem tests careful algebraic manipulation. The key is systematic execution: multiply first to clear fractions, then distribute carefully, and finally isolate the variable while tracking all signs correctly.