Which system of linear equations has no solution?

1. INFER the best solution approach

• To find which system has no solution, I need to test each one using elimination
• A system has no solution when elimination produces a contradiction (like \(0 = 5\))

2. SIMPLIFY each system using elimination

Testing Option A: \(-2\mathrm{x} + 3\mathrm{y} = -9\) and \(2\mathrm{x} - 3\mathrm{y} = 9\)

• Adding directly: \(0 = 0\) → This means infinitely many solutions

Testing Option B: \(2\mathrm{x} - 3\mathrm{y} = 9\) and \(3\mathrm{x} + 4\mathrm{y} = 10\)

• These equations have different slopes → One solution

Testing Option C: \(2\mathrm{x} - 3\mathrm{y} = 9\) and \(-6\mathrm{x} + 9\mathrm{y} = -27\)

• INFER: The second equation looks like a multiple of the first
• SIMPLIFY: Multiply first by -3: \(-6\mathrm{x} + 9\mathrm{y} = -27\)
• This matches the second equation exactly → Infinitely many solutions

Testing Option D: \(-2\mathrm{x} + 3\mathrm{y} = 9\) and \(4\mathrm{x} - 6\mathrm{y} = 18\)

• SIMPLIFY: Multiply first equation by 2: \(-4\mathrm{x} + 6\mathrm{y} = 18\)
• Now I have: \(-4\mathrm{x} + 6\mathrm{y} = 18\) and \(4\mathrm{x} - 6\mathrm{y} = 18\)
• Adding: \(0 = 36\)

3. INFER what the result means

• The equation \(0 = 36\) is never true
• This contradiction means the system has no solution

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing what different elimination results mean

Students successfully perform the elimination steps but don't interpret the results correctly. When they get \(0 = 0\) for options A or C, they might think this means "no solution" because "there's no x or y left." Similarly, when they get \(0 = 36\) for option D, they might not recognize this as a contradiction.

This leads to confusion and random guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors during elimination

Students might incorrectly multiply equations or make sign errors when adding. For example, in option D, they might get \(-4\mathrm{x} + 6\mathrm{y} = 18\) but then incorrectly add it to \(4\mathrm{x} - 6\mathrm{y} = 18\), getting something like \(0 = 0\) instead of \(0 = 36\).

This may lead them to select Choice A or C (thinking they found infinitely many solutions when they should have found no solution).

The Bottom Line:

This problem requires both computational accuracy and conceptual understanding of what elimination results mean. Students need to recognize that contradictions indicate no solution, while \(0 = 0\) indicates infinitely many solutions.