Consider the system of linear equations: 2x - 3y = 5 and 4x - ky = 7, where k is...

1. TRANSLATE the problem requirements

• Given information:
  • Two equations: \(\mathrm{2x - 3y = 5}\) and \(\mathrm{4x - ky = 7}\)
  • System has no solution
  • Need to find value of k

• What "no solution" means: The equations represent parallel but distinct lines

2. INFER the mathematical strategy

• For parallel but distinct lines, we need:
  • Equal slopes (makes them parallel)
  • Different y-intercepts (makes them distinct, not identical)
• This means converting both equations to slope-intercept form \(\mathrm{y = mx + b}\)

3. TRANSLATE both equations to slope-intercept form

First equation: \(\mathrm{2x - 3y = 5}\)

• Move x-term: \(\mathrm{-3y = -2x + 5}\)
• Divide by -3: \(\mathrm{y = \frac{2}{3}x - \frac{5}{3}}\)

Second equation: \(\mathrm{4x - ky = 7}\)

• Move x-term: \(\mathrm{-ky = -4x + 7}\)
• Divide by -k: \(\mathrm{y = \frac{4}{k}x - \frac{7}{k}}\)

4. INFER and set up the parallel condition

• For parallel lines, slopes must be equal:
• Slope of first line = \(\mathrm{\frac{2}{3}}\)
• Slope of second line = \(\mathrm{\frac{4}{k}}\)
• Set equal: \(\mathrm{\frac{2}{3} = \frac{4}{k}}\)

5. SIMPLIFY to solve for k

• Cross multiply: \(\mathrm{2k = 12}\)
• Divide both sides by 2: \(\mathrm{k = 6}\)

6. INFER verification that lines are distinct

• When \(\mathrm{k = 6}\), check y-intercepts:
  • First line: y-intercept = \(\mathrm{-\frac{5}{3}}\)
  • Second line: y-intercept = \(\mathrm{-\frac{7}{6}}\)
• Since \(\mathrm{-\frac{5}{3} \neq -\frac{7}{6}}\), the lines are indeed distinct

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "no solution" with the geometric concept of parallel but distinct lines. They might try to solve the system algebraically by elimination or substitution, get confused when they reach a contradiction like \(\mathrm{0 = -3}\), and then guess randomly rather than understanding what this contradiction means geometrically.

This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify they need equal slopes but make algebraic errors when solving \(\mathrm{\frac{2}{3} = \frac{4}{k}}\). Common mistakes include incorrect cross multiplication (getting \(\mathrm{k = 8}\) instead of \(\mathrm{k = 6}\)) or sign errors when converting to slope-intercept form.

This may lead them to select Choice (C) (6) by luck, or more likely Choice (D) (9) from calculation errors.

The Bottom Line:

This problem requires students to bridge abstract algebra (systems of equations) with geometric intuition (parallel lines). Students who memorize solution techniques without understanding the underlying geometry often struggle to recognize what "no solution" actually means in this context.