In the system of equations below, k is a constant.3x + 2y = 1\(\mathrm{(6 - 2k)x + (4 - k)y...

1. INFER the key condition for infinitely many solutions

When a system of linear equations has infinitely many solutions, it means the two equations actually represent the same line. This happens when one equation is just a scalar multiple of the other.

2. TRANSLATE this condition into mathematical relationships

• Given equations:
  • \(\mathrm{3x + 2y = 1}\)
  • \(\mathrm{(6 - 2k)x + (4 - k)y = 2 - k}\)
• For the second equation to be a scalar multiple of the first:
  • There must exist some number s where: \(\mathrm{s(3x + 2y) = (6 - 2k)x + (4 - k)y}\)
  • And: \(\mathrm{s(1) = 2 - k}\)

3. SIMPLIFY to find the scalar relationship

From the constant terms: \(\mathrm{s = 2 - k}\)

4. SIMPLIFY using coefficient comparison

From the y-coefficients: \(\mathrm{2s = 4 - k}\)

Substitute \(\mathrm{s = 2 - k}\):
\(\mathrm{2(2 - k) = 4 - k}\)
\(\mathrm{4 - 2k = 4 - k}\)
\(\mathrm{-2k = -k}\)
\(\mathrm{-k = 0}\)
\(\mathrm{k = 0}\)

5. INFER verification step

Check with x-coefficients: If \(\mathrm{k = 0}\), then \(\mathrm{s = 2}\)

• \(\mathrm{3s = 3(2) = 6}\)
• \(\mathrm{6 - 2k = 6 - 2(0) = 6}\) ✓

Answer: C (\(\mathrm{k = 0}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the fundamental condition for infinitely many solutions. They might try to solve the system directly or look for when the system has no solution instead.

Without understanding that infinitely many solutions means the equations are identical (scalar multiples), students get confused about what algebraic approach to take. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the scalar multiple relationship correctly but make algebraic errors when solving for k, such as incorrectly distributing or combining like terms.

For example, when solving \(\mathrm{2(2 - k) = 4 - k}\), they might incorrectly get \(\mathrm{4 - 2k = 4 - k}\) and conclude that all values of k work, or make sign errors. This may lead them to select Choice A (-2) or Choice B (-1).

The Bottom Line:

This problem requires understanding the geometric meaning of 'infinitely many solutions' - that the two equations represent the same line. Once students grasp this concept, the algebraic work becomes straightforward coefficient matching.