What is the solution (x, y) to the given system of equations?4x - 3y = 5x = 8

1. INFER the solving strategy

• Given information:
  • \(\mathrm{4x - 3y = 5}\) (first equation)
  • \(\mathrm{x = 8}\) (second equation)
• Since the second equation directly gives us \(\mathrm{x = 8}\), we should substitute this value into the first equation to find y

2. SIMPLIFY through substitution and algebra

• Substitute \(\mathrm{x = 8}\) into the first equation:
  \(\mathrm{4(8) - 3y = 5}\)
• Multiply:
  \(\mathrm{32 - 3y = 5}\)
• Subtract 32 from both sides:
  \(\mathrm{-3y = 5 - 32 = -27}\)
• Divide by -3:
  \(\mathrm{y = \frac{-27}{-3} = 9}\)

3. Write the solution

• The solution is the ordered pair \(\mathrm{(x, y) = (8, 9)}\)

Answer: A. \(\mathrm{(8, 9)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when working with negative coefficients and negative values. They might correctly get to \(\mathrm{-3y = -27}\) but then calculate \(\mathrm{y = -9}\) instead of \(\mathrm{y = 9}\), forgetting that dividing two negative numbers gives a positive result.

This leads them to select Choice C. \(\mathrm{(8, -9)}\)

Second Most Common Error:

Poor INFER reasoning: Some students don't recognize the straightforward substitution approach and instead try to solve the system using elimination or get confused about which variable to solve for first. This can lead to calculation errors or mixing up the order of operations.

This causes them to get stuck and guess among the remaining choices.

The Bottom Line:

This problem tests whether students can execute the substitution method cleanly, particularly handling negative signs correctly during algebraic manipulation. The key insight is recognizing that when one variable is already isolated, substitution becomes the most direct path to solution.