y = 9x + 12 x + 7y = 20 The solution to the given system of equations is \(\mathrm{(x,...

1. TRANSLATE the problem information

• Given system of equations:
  • \(\mathrm{y = 9x + 12}\)
  • \(\mathrm{x + 7y = 20}\)
• Find: The value of y

2. INFER the solution strategy

• Notice that the first equation already has y isolated
• This makes substitution the most efficient method
• Plan: Substitute the expression for y from equation 1 into equation 2

3. SIMPLIFY through substitution

• Substitute \(\mathrm{y = 9x + 12}\) into \(\mathrm{x + 7y = 20}\):
  \(\mathrm{x + 7(9x + 12) = 20}\)
• Distribute carefully:
  \(\mathrm{x + 63x + 84 = 20}\)
• Combine like terms:
  \(\mathrm{64x + 84 = 20}\)
• Solve for x:
  \(\mathrm{64x = -64}\), so \(\mathrm{x = -1}\)

4. SIMPLIFY to find the final answer

• Substitute \(\mathrm{x = -1}\) back into \(\mathrm{y = 9x + 12}\):
  \(\mathrm{y = 9(-1) + 12}\)
  \(\mathrm{= -9 + 12}\)
  \(\mathrm{= 3}\)

Answer: 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor SIMPLIFY execution: Students make arithmetic errors when distributing \(\mathrm{7(9x + 12)}\), often getting \(\mathrm{7(9x) + 12 = 63x + 12}\) instead of \(\mathrm{63x + 84}\).

This leads to the wrong equation \(\mathrm{64x + 12 = 20}\), giving \(\mathrm{x = 1/8}\), and then \(\mathrm{y = 9(1/8) + 12 = 105/8 \approx 13.125}\). This leads to confusion and guessing since such decimal values rarely appear in answer choices.

Second Most Common Error:

Incomplete solution process: Students solve for x correctly but forget to substitute back to find y, stopping at \(\mathrm{x = -1}\).

Since the question asks specifically for the value of y, this causes them to get stuck and randomly select an answer.

The Bottom Line:

While identifying substitution as the strategy is straightforward, success requires careful arithmetic execution through multiple algebraic steps, especially during distribution and when substituting negative values.