y = 6x + 16-{7x - y = 36}What is the solution \(\mathrm{(x, y)}\) to the given system of equations?

1. TRANSLATE the problem information

• Given system of equations:
  • \(\mathrm{y = 6x + 16}\) (Equation 1)
  • \(\mathrm{-7x - y = 36}\) (Equation 2)
• Find: The solution (x, y) that satisfies both equations

2. INFER the most efficient solution method

• Notice that y is already isolated in the first equation
• This makes substitution the ideal method rather than elimination
• We can substitute the expression for y directly into the second equation

3. SIMPLIFY by substituting and solving for x

• Substitute \(\mathrm{y = 6x + 16}\) into \(\mathrm{-7x - y = 36}\):
  \(\mathrm{-7x - (6x + 16) = 36}\)
• Distribute the negative sign:
  \(\mathrm{-7x - 6x - 16 = 36}\)
• Combine like terms:
  \(\mathrm{-13x - 16 = 36}\)
• Add 16 to both sides:
  \(\mathrm{-13x = 52}\)
• Divide by -13:
  \(\mathrm{x = -4}\)

4. SIMPLIFY by finding y

• Substitute \(\mathrm{x = -4}\) back into \(\mathrm{y = 6x + 16}\):
  \(\mathrm{y = 6(-4) + 16}\)
  \(\mathrm{y = -24 + 16}\)
  \(\mathrm{y = -8}\)

Answer: A. (-4, -8)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when distributing the negative sign in \(\mathrm{-7x - (6x + 16) = 36}\).

Instead of correctly getting \(\mathrm{-7x - 6x - 16 = 36}\), they might write \(\mathrm{-7x + 6x + 16 = 36}\) or \(\mathrm{-7x - 6x + 16 = 36}\). This leads to completely different values for x, and subsequently wrong values for y. Depending on the specific error, this may lead them to select Choice B, C, or D rather than the correct answer.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up the substitution but make arithmetic errors when solving \(\mathrm{-13x = 52}\).

They might incorrectly calculate \(\mathrm{x = 4}\) instead of \(\mathrm{x = -4}\), or make errors in the division. When \(\mathrm{x = 4}\), substituting back gives \(\mathrm{y = 6(4) + 16 = 40}\), leading them to select Choice C (4, 40).

The Bottom Line:

This problem tests whether students can execute the substitution method cleanly without making sign or arithmetic errors. The algebraic manipulation involves several steps where small mistakes compound into completely wrong answers.