The point \(\mathrm{(x, 53)}\) is a solution to the system of inequalities in the xy-plane. Which of the following could...

1. TRANSLATE the problem information

• Given information:
  • Point \(\mathrm{(x, 53)}\) is a solution to the system
  • System: \(\mathrm{y \gt 14}\) and \(\mathrm{4x + y \lt 18}\)
• What this tells us: Since \(\mathrm{(x, 53)}\) is a solution, when we substitute x and \(\mathrm{y = 53}\) into both inequalities, both must be true

2. TRANSLATE by substituting into the first inequality

• Substitute \(\mathrm{y = 53}\) into \(\mathrm{y \gt 14}\):
  \(\mathrm{53 \gt 14}\) ✓
• This is automatically satisfied, so we focus on the second inequality

3. SIMPLIFY by substituting and solving the second inequality

• Substitute \(\mathrm{y = 53}\) into \(\mathrm{4x + y \lt 18}\):
  \(\mathrm{4x + 53 \lt 18}\)
• Subtract 53 from both sides:
  \(\mathrm{4x \lt 18 - 53}\)
  \(\mathrm{4x \lt -35}\)
• Divide both sides by 4:
  \(\mathrm{x \lt -8.75}\)

4. APPLY CONSTRAINTS to select the correct answer

• We need \(\mathrm{x \lt -8.75}\)
• Check each choice:
  • A. -9: Is \(\mathrm{-9 \lt -8.75}\)? Yes! ✓
  • B. -5: Is \(\mathrm{-5 \lt -8.75}\)? No
  • C. 5: Is \(\mathrm{5 \lt -8.75}\)? No
  • D. 9: Is \(\mathrm{9 \lt -8.75}\)? No

Answer: A. -9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not understand what "point \(\mathrm{(x, 53)}\) is a solution to the system" means mathematically. They might think they need to find where the inequalities intersect graphically instead of recognizing they need to substitute the coordinates.

This leads to confusion and random guessing rather than systematic substitution.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly substitute \(\mathrm{y = 53}\) but make algebraic errors when solving \(\mathrm{4x + 53 \lt 18}\). Common mistakes include:

• Forgetting to change the inequality direction (incorrectly thinking division affects direction)
• Arithmetic errors: \(\mathrm{18 - 53 = -25}\) instead of \(\mathrm{-35}\)
• Getting \(\mathrm{x \lt -8.75}\) but then incorrectly thinking -5 is less than -8.75

This may lead them to select Choice B (-5) by misunderstanding negative number ordering.

The Bottom Line:

This problem tests whether students can connect the abstract concept of "solution to a system" with the concrete action of substitution, combined with careful inequality manipulation and negative number comparison.