y - 9x = 13 5x = 2y What is the solution \(\mathrm{(x, y)}\) to the given system of equations?...

1. TRANSLATE the problem information

• Given system of equations:
  • \(\mathrm{y - 9x = 13}\)
  • \(\mathrm{5x = 2y}\)
• Need to find: the solution (x, y) that satisfies both equations

2. INFER the best solution approach

• Since the first equation can easily be solved for y, substitution method is ideal
• Strategy: Isolate y from first equation, then substitute into second equation

3. SIMPLIFY to isolate y in the first equation

• Starting with: \(\mathrm{y - 9x = 13}\)
• Add 9x to both sides: \(\mathrm{y = 9x + 13}\)

4. SIMPLIFY by substituting into the second equation

• Replace y in the second equation: \(\mathrm{5x = 2(9x + 13)}\)
• Distribute the 2: \(\mathrm{5x = 18x + 26}\)
• Subtract 18x from both sides: \(\mathrm{5x - 18x = 26}\)
• Combine like terms: \(\mathrm{-13x = 26}\)
• Divide by -13: \(\mathrm{x = -2}\)

5. SIMPLIFY to find y using substitution

• Substitute \(\mathrm{x = -2}\) into \(\mathrm{y = 9x + 13}\)
• \(\mathrm{y = 9(-2) + 13 = -18 + 13 = -5}\)

6. TRANSLATE the final answer format

• The solution is the ordered pair: \(\mathrm{(-2, -5)}\)

Answer: C. (-2, -5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may try elimination method unnecessarily or get confused about which variable to isolate first, leading to more complex algebraic work than needed. Some students might also make sign errors when manipulating the equations, particularly when dealing with \(\mathrm{-13x = 26}\). This can lead them to get \(\mathrm{x = 2}\) instead of \(\mathrm{x = -2}\), which would give the wrong solution and cause confusion when checking answer choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Students solve correctly for \(\mathrm{x = -2}\) and \(\mathrm{y = -5}\), but then write their final answer as \(\mathrm{(-5, -2)}\) instead of \(\mathrm{(-2, -5)}\), confusing the order of the coordinates. This may lead them to select Choice D (-5, -2) even though their mathematical work was correct.

The Bottom Line:

This problem tests both algebraic manipulation skills and attention to coordinate pair formatting. The key insight is recognizing that substitution is efficient here and being careful with negative signs throughout the solution process.