The equation \(-5(\mathrm{z} - 3) = 20\) holds. What is the value of 2z - 6?

1. TRANSLATE the problem information

• Given equation: \(-5(\mathrm{z} - 3) = 20\)
• Need to find: \(2\mathrm{z} - 6\)

2. INFER the most efficient approach

• Key insight: Notice that \(2\mathrm{z} - 6 = 2(\mathrm{z} - 3)\)
• This means we can solve for \((\mathrm{z} - 3)\) directly rather than finding z first
• Strategy: Isolate \((\mathrm{z} - 3)\), then multiply by 2

3. SIMPLIFY to solve for (z - 3)

• Divide both sides by -5:
  \(-5(\mathrm{z} - 3) = 20\)
  \((\mathrm{z} - 3) = 20/(-5)\)
  \((\mathrm{z} - 3) = -4\)

4. SIMPLIFY to find the target expression

• Since \(2\mathrm{z} - 6 = 2(\mathrm{z} - 3)\):
  \(2\mathrm{z} - 6 = 2(-4) = -8\)

Answer: B (-8)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the factoring relationship \(2\mathrm{z} - 6 = 2(\mathrm{z} - 3)\) and instead solve for z completely, then make arithmetic errors.

They correctly get \(\mathrm{z} - 3 = -4\), so \(\mathrm{z} = -1\), but then misread the target expression as \(2(\mathrm{z} - 6)\) instead of \(2\mathrm{z} - 6\). This gives them \(2(-1 - 6) = 2(-7) = -14\).

This may lead them to select Choice A (-14).

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when dividing by the negative coefficient.

They incorrectly solve \(-5(\mathrm{z} - 3) = 20\) as \((\mathrm{z} - 3) = 4\) instead of \((\mathrm{z} - 3) = -4\). Then using the direct method: \(2\mathrm{z} - 6 = 2(\mathrm{z} - 3) = 2(4) = 8\).

This may lead them to select Choice D (8).

The Bottom Line:

This problem rewards students who can spot the algebraic structure and avoid unnecessary steps. The key insight is recognizing that the target expression \(2\mathrm{z} - 6\) can be rewritten as \(2(\mathrm{z} - 3)\), making the solution much more direct.