y = 3x + 9 3y = 8x - 6 The solution to the given system of equations is \(\mathrm{(x,...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{y = 3x + 9}\) (equation 1)
  • \(\mathrm{3y = 8x - 6}\) (equation 2)
• Need to find: \(\mathrm{x - y}\)

2. INFER the solution strategy

• Since equation 1 already expresses y in terms of x, substitution is the most efficient method
• We'll substitute the expression for y from equation 1 into equation 2

3. SIMPLIFY by substitution and algebraic manipulation

• Substitute \(\mathrm{y = 3x + 9}\) into equation 2:
  \(\mathrm{3(3x + 9) = 8x - 6}\)
• Apply distributive property:
  \(\mathrm{9x + 27 = 8x - 6}\)
• Combine like terms:
  \(\mathrm{9x - 8x = -6 - 27}\)
  \(\mathrm{x = -33}\)

4. SIMPLIFY to find y

• Substitute \(\mathrm{x = -33}\) into equation 1:
  \(\mathrm{y = 3(-33) + 9 = -99 + 9 = -90}\)

5. INFER the final calculation

• The question asks for \(\mathrm{x - y}\):
  \(\mathrm{x - y = -33 - (-90) = -33 + 90 = 57}\)

Answer: 57 (Choice D)

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor INFER reasoning: Students solve correctly for \(\mathrm{x = -33}\) and \(\mathrm{y = -90}\), but then calculate \(\mathrm{x + y}\) instead of \(\mathrm{x - y}\) due to misreading the final question.

This gives them \(\mathrm{-33 + (-90) = -123}\), leading them to select Choice A (-123).

Second Most Common Error:

Incomplete solution: Students correctly find \(\mathrm{x = -33}\) but stop there, not realizing they need to find y first before calculating \(\mathrm{x - y}\).

This may lead them to select Choice B (-33), thinking the answer is just the x-value.

Third Most Common Error:

Weak SIMPLIFY execution: Students make sign errors when calculating \(\mathrm{x - y = -33 - (-90)}\), either forgetting that subtracting a negative becomes addition, or making arithmetic mistakes in the final step.

This leads to confusion and potentially selecting Choice C (3) or guessing.

The Bottom Line:

This problem tests both systematic algebraic solving and careful attention to what the question is actually asking for. Students must complete the entire solution process and perform the final calculation correctly.