y gt 10y - 3x lt 7The point \(\mathrm{(x, 25)}\) is a solution to the system of inequalities in the...

1. TRANSLATE the problem information

• Given information:
  • System: \(\mathrm{y \gt 10}\) and \(\mathrm{y - 3x \lt 7}\)
  • Point \(\mathrm{(x, 25)}\) is a solution to this system
• What this means: The point \(\mathrm{(x, 25)}\) must make both inequalities true when we substitute the coordinates

2. APPLY CONSTRAINTS to the first inequality

• Substitute \(\mathrm{y = 25}\) into \(\mathrm{y \gt 10}\):
• \(\mathrm{25 \gt 10}\) ✓
• This is always true, regardless of x value

3. SIMPLIFY the second inequality

• Substitute \(\mathrm{y = 25}\) into \(\mathrm{y - 3x \lt 7}\):
• \(\mathrm{25 - 3x \lt 7}\)
• Subtract 25 from both sides: \(\mathrm{-3x \lt -18}\)
• Divide both sides by -3: \(\mathrm{x \gt 6}\)
• Key: When dividing by a negative number, flip the inequality sign!

4. APPLY CONSTRAINTS to eliminate answer choices

• We need \(\mathrm{x \gt 6}\)
• Check each choice:
  • (A) -9: Is \(\mathrm{-9 \gt 6}\)? No ✗
  • (B) -5: Is \(\mathrm{-5 \gt 6}\)? No ✗
  • (C) 5: Is \(\mathrm{5 \gt 6}\)? No ✗
  • (D) 9: Is \(\mathrm{9 \gt 6}\)? Yes ✓

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Forgetting to reverse the inequality sign when dividing by -3

Students correctly get to \(\mathrm{-3x \lt -18}\), but then divide by -3 without flipping the sign, getting \(\mathrm{x \lt 6}\) instead of \(\mathrm{x \gt 6}\). With \(\mathrm{x \lt 6}\), they would think choices A, B, or C work, leading to confusion about which specific choice to pick. This leads to guessing among the wrong choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Not understanding that "point \(\mathrm{(x, 25)}\) is a solution" means it must satisfy both inequalities

Some students only check one inequality or don't realize they need to substitute the y-coordinate. They might just look at the choices and guess, or try to work backwards from the answer choices without systematic substitution.

The Bottom Line:

This problem tests whether students can systematically work with systems of inequalities and remember the crucial rule about flipping inequality signs. The algebraic manipulation is straightforward, but that one sign-flipping rule is where many students stumble.