Question:2x + 3y = 74x + ky = 5In the system of equations above, k is a constant. For which...

1. INFER the condition for no solution

• For a system of linear equations to have no solution, the lines must be:
  • Parallel (same slope/coefficient ratios)
  • But distinct (not identical)
• In our system \(\mathrm{2x + 3y = 7}\) and \(\mathrm{4x + ky = 5}\), this means:
  • Coefficient ratios must be equal: \(\mathrm{\frac{2}{4} = \frac{3}{k}}\)
  • But the equations can't be scalar multiples of each other

2. SIMPLIFY to find k

• Set up the proportion: \(\mathrm{\frac{2}{4} = \frac{3}{k}}\)
• Cross multiply: \(\mathrm{2k = 12}\)
• Solve: \(\mathrm{k = 6}\)

3. INFER verification strategy

• We need to confirm \(\mathrm{k = 6}\) actually gives no solution (not infinite solutions)
• Check if the equations become inconsistent when \(\mathrm{k = 6}\)

4. SIMPLIFY the verification

• When \(\mathrm{k = 6}\), equation 2 becomes: \(\mathrm{4x + 6y = 5}\)
• Multiply equation 1 by 2: \(\mathrm{4x + 6y = 14}\)
• Result: \(\mathrm{4x + 6y = 14}\) and \(\mathrm{4x + 6y = 5}\)
• Since \(\mathrm{14 \neq 5}\), the system is inconsistent (no solution)

Answer: D (6)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the conditions for no solution vs infinite solutions. They correctly find \(\mathrm{k = 6}\) makes the lines parallel, but don't verify the lines are distinct rather than identical.

Without checking that \(\mathrm{4x + 6y = 14}\) contradicts \(\mathrm{4x + 6y = 5}\), they might think \(\mathrm{k = 6}\) gives infinite solutions instead of no solution. This confusion about which condition applies leads to random guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the wrong proportion or make arithmetic errors. They might write \(\mathrm{\frac{2}{4} = \frac{k}{3}}\) instead of \(\mathrm{\frac{2}{4} = \frac{3}{k}}\), leading to \(\mathrm{k = \frac{3}{2}}\) or similar incorrect values.

This may lead them to select Choice C (3) by incorrectly solving \(\mathrm{\frac{k}{3} = \frac{1}{2}}\).

The Bottom Line:

The key insight is distinguishing between parallel identical lines (infinite solutions) and parallel distinct lines (no solution). Students must both find the parallel condition AND verify the equations are inconsistent.