In the xy-plane, a system of two linear equations has no solution. If the first equation is 3x - y...

1. TRANSLATE the problem information

• Given information:
  • First equation: \(\mathrm{3x - y = -5}\)
  • Second equation: \(\mathrm{y = kx + 2}\)
  • The system has no solution
• We need to find the value of constant k

2. INFER the key mathematical relationship

• "No solution" means the lines are parallel
• Parallel lines have the same slope but different y-intercepts
• Strategy: Find both slopes and set them equal

3. SIMPLIFY the first equation to slope-intercept form

• Convert \(\mathrm{3x - y = -5}\) to \(\mathrm{y = mx + b}\) form:
  \(\mathrm{3x - y = -5}\)
  \(\mathrm{-y = -3x - 5}\)
  \(\mathrm{y = 3x + 5}\)
• The first line has slope 3 and y-intercept 5

4. INFER the slope relationship

• The second equation \(\mathrm{y = kx + 2}\) already shows slope k and y-intercept 2
• For parallel lines: slope of first line = slope of second line
• Therefore: \(\mathrm{3 = k}\), so \(\mathrm{k = 3}\)

5. Verify the no-solution condition

• Line 1: \(\mathrm{y = 3x + 5}\) (slope = 3, y-intercept = 5)
• Line 2: \(\mathrm{y = 3x + 2}\) (slope = 3, y-intercept = 2)
• Same slope ✓, different y-intercepts ✓ → No solution confirmed

Answer: C. 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "no solution" to the parallel lines condition. They might try to solve the system algebraically by substitution or elimination, getting confused when they reach a contradiction like \(\mathrm{5 = 2}\). This leads to confusion and guessing rather than recognizing this contradiction actually confirms no solution exists.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make algebraic errors when converting \(\mathrm{3x - y = -5}\) to slope-intercept form, such as getting \(\mathrm{y = -3x + 5}\) (wrong slope) or \(\mathrm{y = 3x - 5}\) (wrong y-intercept). This incorrect slope leads them to select Choice A (-3) as the answer.

The Bottom Line:

This problem tests whether students understand that systems of equations can be analyzed geometrically through slope relationships, not just solved algebraically. The key insight is translating "no solution" into the parallel lines condition.