Question:9x + 6y = 15k/7x + 2y = 10In the given system of equations, k is a constant. If the...

1. INFER what "no solution" means geometrically

• A system has no solution when the lines are parallel but not identical
• This means: same slope, different y-intercepts
• Strategy: Convert both equations to slope-intercept form, then set slopes equal

2. SIMPLIFY the first equation to slope-intercept form

• Start with: \(9\mathrm{x} + 6\mathrm{y} = 15\)
• Subtract \(9\mathrm{x}\): \(6\mathrm{y} = -9\mathrm{x} + 15\)
• Divide by 6: \(\mathrm{y} = \frac{-9}{6}\mathrm{x} + \frac{15}{6}\)
• Reduce fractions: \(\mathrm{y} = \frac{-3}{2}\mathrm{x} + \frac{5}{2}\)
• Slope of first line = \(\frac{-3}{2}\)

3. SIMPLIFY the second equation to slope-intercept form

• Start with: \(\frac{\mathrm{k}}{7}\mathrm{x} + 2\mathrm{y} = 10\)
• Subtract \(\frac{\mathrm{k}}{7}\mathrm{x}\): \(2\mathrm{y} = -\frac{\mathrm{k}}{7}\mathrm{x} + 10\)
• Divide by 2: \(\mathrm{y} = -\frac{\mathrm{k}}{14}\mathrm{x} + 5\)
• Slope of second line = \(-\frac{\mathrm{k}}{14}\)

4. SIMPLIFY by setting slopes equal for parallel lines

• For parallel lines: \(\frac{-3}{2} = \frac{-\mathrm{k}}{14}\)
• Multiply both sides by -1: \(\frac{3}{2} = \frac{\mathrm{k}}{14}\)
• Multiply both sides by 14: \(\mathrm{k} = \frac{3}{2} \times 14 = 21\)

5. INFER verification that lines are truly parallel but not identical

• When \(\mathrm{k} = 21\), first line: \(\mathrm{y} = \frac{-3}{2}\mathrm{x} + \frac{5}{2}\), so y-intercept = 2.5
• Second line: \(\mathrm{y} = \frac{-3}{2}\mathrm{x} + 5\), so y-intercept = 5
• Since \(2.5 ≠ 5\), lines are parallel but distinct → no solution confirmed

Answer: D) 21

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "no solution" to the geometric condition of parallel lines. Instead, they might try to solve the system directly by elimination or substitution, get confused when variables cancel to something like "\(0 = 5\)," and then guess randomly rather than understanding this confirms no solution.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic mistakes when converting to slope-intercept form or solving \(\frac{-3}{2} = \frac{-\mathrm{k}}{14}\). Common errors include sign mistakes, fraction errors, or incorrectly handling the \(\frac{\mathrm{k}}{7}\) term, leading to wrong values like \(\mathrm{k} = -21\) or \(\mathrm{k} = 42\).

This may lead them to select Choice A (-21) or Choice E (42).

The Bottom Line:

This problem requires both conceptual understanding (what "no solution" means geometrically) and solid algebraic manipulation skills. Students who miss either component will struggle to reach the correct answer systematically.