Question:2/3x + 1/2y = 5/6 + 1/3x3/4y - 1/2 = kx + 1/4yIn the given system of equations, k is...

1. TRANSLATE the problem information

• Given: System of equations with parameter k
• Find: Value of k that makes system have no solution
• Key insight: "No solution" is a specific mathematical condition

2. SIMPLIFY both equations to standard form

• First equation: \(\frac{2}{3}\mathrm{x} + \frac{1}{2}\mathrm{y} = \frac{5}{6} + \frac{1}{3}\mathrm{x}\)
  • Move x terms to left: \((\frac{2}{3} - \frac{1}{3})\mathrm{x} + \frac{1}{2}\mathrm{y} = \frac{5}{6}\)
  • Result: \(\frac{1}{3}\mathrm{x} + \frac{1}{2}\mathrm{y} = \frac{5}{6}\)
• Second equation: \(\frac{3}{4}\mathrm{y} - \frac{1}{2} = \mathrm{kx} + \frac{1}{4}\mathrm{y}\)
  • Move y terms to left: \((\frac{3}{4} - \frac{1}{4})\mathrm{y} = \mathrm{kx} + \frac{1}{2}\)
  • Simplify: \(\frac{1}{2}\mathrm{y} = \mathrm{kx} + \frac{1}{2}\)
  • Rearrange: \(-\mathrm{kx} + \frac{1}{2}\mathrm{y} = \frac{1}{2}\)

3. INFER the condition for no solution

• No solution occurs when equations represent parallel lines
• Parallel lines: same slope, different y-intercepts
• Need to convert to slope-intercept form: \(\mathrm{y} = \mathrm{mx} + \mathrm{b}\)

4. SIMPLIFY to slope-intercept form

• Equation 1: \(\frac{1}{3}\mathrm{x} + \frac{1}{2}\mathrm{y} = \frac{5}{6}\)
  • Solve for y: \(\frac{1}{2}\mathrm{y} = -\frac{1}{3}\mathrm{x} + \frac{5}{6}\)
  • Multiply by 2: \(\mathrm{y} = -\frac{2}{3}\mathrm{x} + \frac{5}{3}\)
• Equation 2: \(-\mathrm{kx} + \frac{1}{2}\mathrm{y} = \frac{1}{2}\)
  • Solve for y: \(\frac{1}{2}\mathrm{y} = \mathrm{kx} + \frac{1}{2}\)
  • Multiply by 2: \(\mathrm{y} = 2\mathrm{kx} + 1\)

5. APPLY CONSTRAINTS for parallel lines

• For same slope: \(-\frac{2}{3} = 2\mathrm{k}\)
• Solve: \(\mathrm{k} = -\frac{2}{3} \div 2 = -\frac{1}{3}\)
• Verify different y-intercepts: \(\frac{5}{3} \neq 1\) ✓

Answer: D) -1/3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Making fraction arithmetic errors when collecting like terms or converting between forms.

Students often struggle with operations like \((\frac{2}{3} - \frac{1}{3})\mathrm{x}\) or converting \(\frac{1}{2}\mathrm{y} = -\frac{1}{3}\mathrm{x} + \frac{5}{6}\) to slope-intercept form. A common mistake is getting \(\mathrm{y} = -\frac{1}{3}\mathrm{x} + \frac{5}{6}\) instead of \(\mathrm{y} = -\frac{2}{3}\mathrm{x} + \frac{5}{3}\), leading to the wrong slope comparison. This may lead them to select Choice B (-2/3) or get confused and guess.

Second Most Common Error:

Incomplete INFER reasoning: Understanding that no solution means something special, but not connecting it to the parallel lines condition.

Some students recognize they need equal slopes but forget to verify different y-intercepts, or they set the entire equations equal instead of just the slopes. This leads to confusion and guessing among the available choices.

The Bottom Line:

This problem combines algebraic manipulation with systems theory. Success requires both technical fraction skills and conceptual understanding of what "no solution" means geometrically.