Question:x + 2y = -53x = y + 6What is the solution \(\mathrm{(x, y)}\) to the given system of equations?

1. TRANSLATE the problem information

• Given information:
  • First equation: \(\mathrm{x + 2y = -5}\)
  • Second equation: \(\mathrm{3x = y + 6}\)
• We need to find the values of x and y that satisfy both equations simultaneously

2. INFER the approach

• Since we have two equations with two unknowns, we can use substitution or elimination
• The first equation can be easily solved for x, making substitution efficient
• Strategy: Solve the first equation for x, then substitute into the second equation

3. SIMPLIFY to solve for x from the first equation

• Starting with: \(\mathrm{x + 2y = -5}\)
• Subtract 2y from both sides: \(\mathrm{x = -5 - 2y}\)

4. SIMPLIFY by substituting into the second equation

• Replace x in the second equation: \(\mathrm{3x = y + 6}\)
• Substitute \(\mathrm{x = -5 - 2y}\): \(\mathrm{3(-5 - 2y) = y + 6}\)
• Distribute the 3: \(\mathrm{-15 - 6y = y + 6}\)

5. SIMPLIFY to solve for y

• Move all y terms to one side: \(\mathrm{-15 - 6y - y = 6}\)
• Combine like terms: \(\mathrm{-15 - 7y = 6}\)
• Add 15 to both sides: \(\mathrm{-7y = 21}\)
• Divide by -7: \(\mathrm{y = -3}\)

6. SIMPLIFY to find x

• Substitute \(\mathrm{y = -3}\) back into \(\mathrm{x = -5 - 2y}\)
• \(\mathrm{x = -5 - 2(-3) = -5 + 6 = 1}\)

Answer: \(\mathrm{(1, -3)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign errors when distributing or combining terms

When substituting \(\mathrm{3(-5 - 2y) = y + 6}\), students often make errors like:

• Distributing incorrectly: \(\mathrm{3(-5 - 2y) = -15 + 6y}\) (forgetting the negative sign on the 2y term)
• Or when moving terms: writing \(\mathrm{-15 - 6y = y + 6}\) as \(\mathrm{-15 - 6y + y = 6}\) (missing the sign change)

These algebraic mistakes lead to incorrect values for y, which then produces wrong values for x. This may lead them to select Choice A \(\mathrm{(-3, 1)}\) or Choice D \(\mathrm{(-1, 3)}\).

Second Most Common Error:

Poor INFER reasoning: Confusing which value corresponds to which variable in the final answer

Students correctly solve for \(\mathrm{y = -3}\) and \(\mathrm{x = 1}\), but then write the answer as \(\mathrm{(-3, 1)}\) instead of \(\mathrm{(1, -3)}\), forgetting that coordinate pairs are written as \(\mathrm{(x, y)}\). This may lead them to select Choice A \(\mathrm{(-3, 1)}\).

The Bottom Line:

This problem tests both systematic algebraic manipulation skills and careful attention to coordinate notation. Success requires methodical execution of multi-step algebra while maintaining accuracy with signs and variable labels.