\((\mathrm{k}-2)\mathrm{x} - 5\mathrm{y} = 7\)3x - 2y = 4In the given system of equations, k is a constant. If the...

1. INFER the meaning of "no solution"

• When a system has no solution, the lines are parallel but not identical
• This means: same slope, different y-intercepts
• Strategy: Convert both equations to \(\mathrm{y = mx + b}\) form to compare slopes and y-intercepts

2. SIMPLIFY the first equation to slope-intercept form

• Starting with: \(\mathrm{(k-2)x - 5y = 7}\)
• Isolate y: \(\mathrm{-5y = -(k-2)x + 7}\)
• Divide by -5: \(\mathrm{y = \frac{k-2}{5}x - \frac{7}{5}}\)

Key insight: Slope \(\mathrm{m_1 = \frac{k-2}{5}}\), y-intercept = \(\mathrm{-\frac{7}{5}}\)

3. SIMPLIFY the second equation to slope-intercept form

• Starting with: \(\mathrm{3x - 2y = 4}\)
• Isolate y: \(\mathrm{-2y = -3x + 4}\)
• Divide by -2: \(\mathrm{y = \frac{3}{2}x - 2}\)

Key insight: Slope \(\mathrm{m_2 = \frac{3}{2}}\), y-intercept = \(\mathrm{-2}\)

4. INFER the condition for parallel lines and solve for k

• For parallel lines: \(\mathrm{m_1 = m_2}\)
• Set up equation: \(\mathrm{\frac{k-2}{5} = \frac{3}{2}}\)
• SIMPLIFY: Cross-multiply: \(\mathrm{2(k-2) = 15}\)
• Expand: \(\mathrm{2k - 4 = 15}\)
• Solve: \(\mathrm{2k = 19}\), so \(\mathrm{k = 9.5}\)

5. INFER verification of "no solution" condition

• Check y-intercepts are different:
  • First line: \(\mathrm{-\frac{7}{5} = -1.4}\)
  • Second line: \(\mathrm{-2}\)
• Since \(\mathrm{-1.4 \neq -2}\), lines are parallel but distinct ✓

Answer: C) 9.5

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 think no solution means the slopes should be different, leading them to set up \(\mathrm{\frac{k-2}{5} \neq \frac{3}{2}}\) or get confused about what equation to solve. This leads to confusion and abandoning the systematic approach, causing them to guess randomly.

Second Most Common Error:

Incomplete INFER reasoning: Students correctly set slopes equal but forget to verify that y-intercepts are different. They solve for \(\mathrm{k = 9.5}\) but don't confirm this actually creates a no-solution system. While this might still lead to the correct answer, they miss the complete logical reasoning and might doubt their work.

The Bottom Line:

This problem tests whether students understand the geometric meaning behind algebraic conditions for systems of equations. The key insight is recognizing that "no solution" translates to a specific mathematical relationship between the coefficients.