Which system of linear equations has no solution?2x + 3y = 12y = -2/3x + 42x + 3y = 12y...

1. INFER the key strategy

• To determine which system has no solution, we need to test each system
• The most efficient approach: use substitution since each system has one equation already solved for y
• A system has no solution when we get a contradiction (like 5 = 8)

2. SIMPLIFY each system through substitution

Test Choice A: \(\mathrm{2x + 3y = 12}\) and \(\mathrm{y = -\frac{2}{3}x + 4}\)

• Substitute the second equation into the first:
  \(\mathrm{2x + 3(-\frac{2}{3}x + 4) = 12}\)
• SIMPLIFY: \(\mathrm{2x - 2x + 12 = 12}\)
• Result: \(\mathrm{12 = 12}\) ✓

3. INFER what this result means

• When we get \(\mathrm{12 = 12}\) (always true), the system has infinite solutions
• This happens when both equations represent the same line

4. SIMPLIFY Choice B: \(\mathrm{2x + 3y = 12}\) and \(\mathrm{y = -\frac{2}{3}x + 6}\)

• Substitute: \(\mathrm{2x + 3(-\frac{2}{3}x + 6) = 12}\)
• SIMPLIFY: \(\mathrm{2x - 2x + 18 = 12}\)
• Result: \(\mathrm{18 = 12}\)

5. INFER the contradiction

• \(\mathrm{18 = 12}\) is impossible - this is a contradiction!
• When substitution leads to a false statement, the system has no solution
• This means the lines are parallel (same slope, different y-intercepts)

6. Verify this is the answer by checking the remaining choices

Choice C: Results in \(\mathrm{4x = 0}\), so \(\mathrm{x = 0}\) and \(\mathrm{y = 4}\) (one unique solution)
Choice D: The second equation is exactly 2 × the first equation, so infinite solutions

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize what different algebraic results mean for the system's solutions. They might correctly perform the substitution and get "\(\mathrm{18 = 12}\)" but fail to interpret this as indicating no solution. Some students think this means they made an algebra error and restart the problem, while others might select a different choice thinking their work was wrong.

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

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors during substitution, particularly with the distributive property: \(\mathrm{3(-\frac{2}{3}x + 6)}\). They might calculate \(\mathrm{3(-\frac{2}{3}x)}\) incorrectly as \(\mathrm{-6x}\) instead of \(\mathrm{-2x}\), or handle the signs incorrectly when distributing. This leads them to get different contradictions or even consistent equations.

This may lead them to select Choice A or Choice C depending on their specific calculation error.

The Bottom Line:

This problem tests both algebraic manipulation skills and conceptual understanding of what different outcomes mean in systems of equations. Success requires not just correct calculations, but also the insight to recognize contradictions as indicators of inconsistent systems.