The solution to the given system of equations is \(\mathrm{(x, y)}\). What is the value of y?x = 8x +...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{x = 8}\) (first equation)
  • \(\mathrm{x + 3y = 26}\) (second equation)
• What this tells us: We need to find the value of y that makes both equations true

2. INFER the most efficient approach

• Since x is already isolated in the first equation (\(\mathrm{x = 8}\)), substitution is the natural choice
• We can substitute this known value directly into the second equation

3. SIMPLIFY through substitution and algebraic steps

• Substitute \(\mathrm{x = 8}\) into the second equation:
  \(\mathrm{8 + 3y = 26}\)
• Subtract 8 from both sides:
  \(\mathrm{3y = 18}\)
• Divide both sides by 3:
  \(\mathrm{y = 6}\)

Answer: 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors during the algebraic steps, particularly when subtracting 8 from both sides or dividing by 3.

For example, they might incorrectly calculate \(\mathrm{26 - 8 = 16}\) instead of 18, leading to \(\mathrm{y = 16/3 \approx 5.33}\). This leads to confusion since the answer doesn't match any clean integer value they might expect.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might misinterpret what the problem is asking for and attempt to solve for x instead of y, or they might not recognize that they already have the complete solution after finding \(\mathrm{y = 6}\).

This can cause them to second-guess their work or attempt unnecessary additional steps, potentially introducing errors or leading to confusion about the final answer.

The Bottom Line:

This problem tests whether students can execute a straightforward substitution method without making careless arithmetic errors. The conceptual understanding is minimal, but precision in algebraic manipulation is crucial.