The system of equations below has variables x and y, where k is a constant.2x + 3y = 7kx +...

1. INFER what 'infinitely many solutions' means

• For a system to have infinitely many solutions, the two equations must represent the same line
• This happens when one equation is a scalar multiple of the other
• In other words, all corresponding coefficients must be proportional

2. INFER the solution strategy

• We need the ratios of corresponding coefficients to be equal:
  • coefficient of x in equation 1 / coefficient of x in equation 2 = constant
  • coefficient of y in equation 1 / coefficient of y in equation 2 = same constant
  • constant term in equation 1 / constant term in equation 2 = same constant

3. TRANSLATE this into mathematical ratios

• From our system: \(\mathrm{2x + 3y = 7}\) and \(\mathrm{kx + 6y = 14}\)
• We need: \(\frac{2}{\mathrm{k}} = \frac{3}{6} = \frac{7}{14}\)

4. SIMPLIFY the known ratios first

• \(\frac{3}{6} = \frac{1}{2}\)
• \(\frac{7}{14} = \frac{1}{2}\)
• So we need: \(\frac{2}{\mathrm{k}} = \frac{1}{2}\)

5. SIMPLIFY to solve for k

• Cross multiply: \(\frac{2}{\mathrm{k}} = \frac{1}{2}\)
• \(2 \times 2 = \mathrm{k} \times 1\)
• \(4 = \mathrm{k}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize what 'infinitely many solutions' means for a system of equations. They might think it means the coefficients should be equal (not proportional), leading them to look for k where the x-coefficients match: \(2 = \mathrm{k}\), giving them Choice A (2).

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the correct proportion \(\frac{2}{\mathrm{k}} = \frac{3}{6}\) but make calculation errors. They might incorrectly simplify \(\frac{3}{6}\) as \(\frac{3}{6} = \frac{1}{3}\) instead of \(\frac{1}{2}\), leading to \(\frac{2}{\mathrm{k}} = \frac{1}{3}\), so \(\mathrm{k = 6}\). This causes them to select Choice D (6).

The Bottom Line:

This problem requires understanding the geometric meaning of 'infinitely many solutions' - the equations represent the same line. Students who focus only on algebraic manipulation without this conceptual insight often struggle to set up the correct approach.