Consider the system of equations \((\mathrm{k} + 1)\mathrm{x} - 4\mathrm{y} = 7\) and 3x - 12y = 24. The system...

1. INFER the condition for no solution

• For a system of linear equations to have no solution, the lines must be:
  • Parallel (same slope)
  • Distinct (different y-intercepts)

• This means we need the coefficients of x and y to be proportional, but the constant terms to create different y-intercepts.

2. SIMPLIFY both equations to slope-intercept form

• First equation: \((\mathrm{k} + 1)\mathrm{x} - 4\mathrm{y} = 7\)
  • Isolate y: \(-4\mathrm{y} = -(\mathrm{k} + 1)\mathrm{x} + 7\)
  • Divide by -4: \(\mathrm{y} = \frac{(\mathrm{k} + 1)}{4}\mathrm{x} - \frac{7}{4}\)

• Second equation: \(3\mathrm{x} - 12\mathrm{y} = 24\)
  • Isolate y: \(-12\mathrm{y} = -3\mathrm{x} + 24\)
  • Divide by -12: \(\mathrm{y} = \frac{1}{4}\mathrm{x} - 2\)

3. INFER the parallel condition and solve for k

• For parallel lines, slopes must be equal:
  \(\frac{(\mathrm{k} + 1)}{4} = \frac{1}{4}\)

• SIMPLIFY to solve for k:
  • Multiply both sides by 4: \(\mathrm{k} + 1 = 1\)
  • Subtract 1: \(\mathrm{k} = 0\)

4. INFER verification of the no-solution condition

• When \(\mathrm{k} = 0\):
  • First line: \(\mathrm{y} = \frac{1}{4}\mathrm{x} - \frac{7}{4}\)
  • Second line: \(\mathrm{y} = \frac{1}{4}\mathrm{x} - 2\)

• Same slope \(\left(\frac{1}{4}\right)\) ✓
• Different y-intercepts \(\left(-\frac{7}{4} \neq -2\right)\) ✓
• Therefore: parallel but distinct lines = no solution ✓

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "no solution" with the parallel lines condition. They might try to solve the system directly by elimination or substitution, get contradictory results, and then guess randomly without understanding why the contradiction occurred.

This leads to confusion and guessing rather than systematic analysis.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when converting to slope-intercept form, particularly with the signs or fractions. For example, getting \(\mathrm{y} = \frac{(\mathrm{k} + 1)}{4}\mathrm{x} + \frac{7}{4}\) instead of \(\mathrm{y} = \frac{(\mathrm{k} + 1)}{4}\mathrm{x} - \frac{7}{4}\), which leads to incorrect slope equality setup.

This may lead them to select Choice A (-1) or Choice C (1) based on faulty algebra.

The Bottom Line:

The key insight is recognizing that "no solution" has a specific geometric meaning (parallel but distinct lines), which translates to a specific algebraic condition (equal slopes, different intercepts). Students who miss this connection often get lost in computational attempts rather than strategic analysis.