y = -2x3x + y = 40The solution to the given system of equations is \(\mathrm{(x, y)}\). What is the...

1. INFER the most efficient solution strategy

• Given information:
  • First equation: \(\mathrm{y = -2x}\) (y is already isolated)
  • Second equation: \(\mathrm{3x + y = 40}\)

• Since y is already solved for in the first equation, substitution will be most direct

2. INFER what to substitute where

• We can substitute the expression for y from the first equation into the second equation
• Replace y with (-2x) in the equation \(\mathrm{3x + y = 40}\)

3. SIMPLIFY by substituting and combining like terms

• Substitute: \(\mathrm{3x + (-2x) = 40}\)
• Combine like terms: \(\mathrm{3x - 2x = x}\)
• Therefore: \(\mathrm{x = 40}\)

Answer: 40

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when combining like terms

When substituting \(\mathrm{y = -2x}\), they might write \(\mathrm{3x + (-2x) = 40}\) correctly, but then combine terms incorrectly as \(\mathrm{3x - 2x = 5x}\) instead of x, leading to the wrong equation \(\mathrm{5x = 40}\) and the incorrect answer \(\mathrm{x = 8}\).

This causes confusion since none of the typical answer choices would match this error.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize the straightforward substitution opportunity

Instead of using the already-isolated \(\mathrm{y = -2x}\), they might try to use elimination method or rearrange equations unnecessarily, leading to more complex work and potential arithmetic errors along the way.

This leads to confusion and potential guessing due to overcomplicating a simple problem.

The Bottom Line:

This problem rewards students who can quickly identify when substitution is the obvious choice and execute basic algebraic simplification accurately. The key insight is recognizing that having y already isolated makes this a one-step substitution problem.