3y = 4x + 17-{3y = 9x - 23}The solution to the given system of equations is \((\mathrm{x}, \mathrm{y})\). What...

1. TRANSLATE the problem information

• Given system of equations:
  • \(\mathrm{3y = 4x + 17}\)
  • \(\mathrm{-3y = 9x - 23}\)
• Need to find: the value of \(\mathrm{39x}\)

2. INFER the solution strategy

• Notice that the y-coefficients are opposites: \(\mathrm{+3y}\) and \(\mathrm{-3y}\)
• This means we can add the equations together to eliminate y entirely
• This avoids needing to solve for individual values of x and y

3. SIMPLIFY by adding the equations

Add the left sides and right sides:

• Left side: \(\mathrm{3y + (-3y) = 0}\)
• Right side: \(\mathrm{(4x + 17) + (9x - 23) = 4x + 9x + 17 - 23 = 13x - 6}\)
• Result: \(\mathrm{0 = 13x - 6}\)

4. SIMPLIFY to solve for x

• Add 6 to both sides: \(\mathrm{6 = 13x}\)
• So \(\mathrm{x = \frac{6}{13}}\)

5. SIMPLIFY to find 39x

• \(\mathrm{39x = 39 \times \frac{6}{13}}\)
  \(\mathrm{= \frac{39 \times 6}{13}}\)
  \(\mathrm{= \frac{234}{13}}\)
  \(\mathrm{= 18}\)

Answer: D. 18

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the elimination opportunity and instead try to solve the system using substitution or other complex methods. They might solve for y from the first equation (\(\mathrm{y = \frac{4x + 17}{3}}\)) and substitute into the second equation, creating messy fractions and calculation errors. This leads to confusion and potentially guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the elimination approach but make arithmetic errors when combining like terms or calculating the final answer. For example, they might get \(\mathrm{13x = 6}\) but then incorrectly calculate \(\mathrm{39x}\) by forgetting that \(\mathrm{39 = 3 \times 13}\), leading them to select Choice C (6) instead of multiplying appropriately.

The Bottom Line:

This problem rewards students who can quickly spot the elimination opportunity rather than getting bogged down in complex algebraic manipulations. The key insight is recognizing that the question asks for \(\mathrm{39x}\), not x itself, which makes the elimination method even more efficient.