Question:2x + y leq 6x - 3y geq -8Which point \((\mathrm{x}, \mathrm{y})\) is a solution to the given system of...

1. TRANSLATE the problem requirements

• Given: Two inequalities forming a system:
  • \(\mathrm{2x + y \leq 6}\)
  • \(\mathrm{x - 3y \geq -8}\)
• Four candidate points to test: \(\mathrm{(-4, 2)}\), \(\mathrm{(1, 4)}\), \(\mathrm{(2, 1)}\), \(\mathrm{(5, -1)}\)
• Need to find which point satisfies both inequalities

2. INFER the solution strategy

• A point is a solution to the system only if it satisfies both inequalities
• Must test each point by substituting its coordinates into both inequalities
• If any inequality is violated, that point is not a solution

3. TRANSLATE and SIMPLIFY for each point

Testing A) (-4, 2):

• First inequality: TRANSLATE \(\mathrm{x = -4}\), \(\mathrm{y = 2}\) into \(\mathrm{2x + y \leq 6}\)
  • SIMPLIFY: \(\mathrm{2(-4) + 2 = -8 + 2 = -6}\)
  • Is \(\mathrm{-6 \leq 6}\)? Yes ✓
• Second inequality: TRANSLATE into \(\mathrm{x - 3y \geq -8}\)
  • SIMPLIFY: \(\mathrm{(-4) - 3(2) = -4 - 6 = -10}\)
  • Is \(\mathrm{-10 \geq -8}\)? No ✗

Testing B) (1, 4):

• First inequality: \(\mathrm{2(1) + 4 = 6}\). Is \(\mathrm{6 \leq 6}\)? Yes ✓
• Second inequality: \(\mathrm{1 - 3(4) = 1 - 12 = -11}\). Is \(\mathrm{-11 \geq -8}\)? No ✗

Testing C) (2, 1):

• First inequality: \(\mathrm{2(2) + 1 = 5}\). Is \(\mathrm{5 \leq 6}\)? Yes ✓
• Second inequality: \(\mathrm{2 - 3(1) = 2 - 3 = -1}\). Is \(\mathrm{-1 \geq -8}\)? Yes ✓

Testing D) (5, -1):

• First inequality: \(\mathrm{2(5) + (-1) = 9}\). Is \(\mathrm{9 \leq 6}\)? No ✗

4. INFER the final answer

• Only point C satisfies both inequalities
• Therefore, \(\mathrm{(2, 1)}\) is the solution

Answer: C) (2, 1)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students often test only one inequality or think a point is valid if it satisfies just one inequality, not realizing both must be satisfied simultaneously.

For example, they might see that \(\mathrm{(-4, 2)}\) satisfies the first inequality and immediately select Choice A ((-4, 2)) without checking the second inequality.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when working with negative numbers, especially in expressions like \(\mathrm{(-4) - 3(2)}\) or \(\mathrm{x - 3y}\) when \(\mathrm{y}\) is positive.

A common mistake is calculating \(\mathrm{(-4) - 3(2)}\) as \(\mathrm{(-4) - 6 = -2}\) instead of \(\mathrm{-10}\), leading them to incorrectly validate point A and select Choice A ((-4, 2)).

The Bottom Line:

This problem tests systematic thinking and careful arithmetic with signed numbers. Students who rush or don't verify both conditions often select the first point that appears to work in one inequality.