2x + 3y = 66x + ky = 9In the given system of equations, k is a constant. If the...

1. INFER what no solution means for a system

• A system has no solution when the equations represent parallel lines that never intersect
• This happens when:
  • The coefficients of x and y are proportional (same slope)
  • BUT the constant terms don't follow the same proportion (different y-intercepts)

2. TRANSLATE the given information

• Given system:
  • \(\mathrm{2x + 3y = 6}\)
  • \(\mathrm{6x + ky = 9}\)
• We need to find k that makes this system have no solution

3. INFER the condition for proportional coefficients

• For the lines to be parallel, the ratios of corresponding coefficients must be equal:
  • x-coefficient ratio: \(\mathrm{\frac{2}{6} = \frac{1}{3}}\)
  • y-coefficient ratio: \(\mathrm{\frac{3}{k}}\)
• Therefore: \(\mathrm{\frac{3}{k} = \frac{1}{3}}\)

4. SIMPLIFY to solve for k

• From \(\mathrm{\frac{3}{k} = \frac{1}{3}}\), cross multiply:

\(\mathrm{3 \times 3 = k \times 1}\)

\(\mathrm{9 = k}\)

• So \(\mathrm{k = 9}\)

5. INFER verification of no solution

• With \(\mathrm{k = 9}\), our system becomes:
  • \(\mathrm{2x + 3y = 6}\)
  • \(\mathrm{6x + 9y = 9}\)
• Divide the second equation by 3: \(\mathrm{2x + 3y = 3}\)
• Now we have: \(\mathrm{2x + 3y = 6}\) AND \(\mathrm{2x + 3y = 3}\)
• This is impossible - the same expression cannot equal two different numbers!

Answer: C) 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse no solution with infinitely many solutions

They think both conditions require proportional coefficients, so they set up \(\mathrm{\frac{2}{6} = \frac{3}{k} = \frac{6}{9}}\), leading to \(\mathrm{\frac{3}{k} = \frac{2}{3}}\), which gives \(\mathrm{k = \frac{9}{2} = 4.5}\). Since 4.5 isn't among the choices, this leads to confusion and guessing.

Second Most Common Error:

Incomplete INFER reasoning: Students correctly find \(\mathrm{k = 9}\) but don't verify it actually creates no solution

They might worry they made an error since they didn't check their work, leading them to second-guess and potentially select Choice A (-9) by incorrectly thinking about sign relationships, or Choice B (0) assuming a special case.

The Bottom Line:

The key insight is distinguishing between no solution (parallel lines) and infinitely many solutions (identical lines). No solution requires coefficients to be proportional BUT constants to break that same proportion.