2x + 3y = 12ax + 6y = 18In the system of equations above, a is a constant. If the...

1. TRANSLATE the problem information

• Given information:
  • System: \(\mathrm{2x + 3y = 12}\) and \(\mathrm{ax + 6y = 18}\)
  • The system has no solution
  • Need to find the value of constant a
• What "no solution" tells us: The equations represent parallel lines that never intersect

2. INFER the mathematical condition

• For no solution, lines must be parallel but not identical
• Parallel lines have the same slope, which means proportional coefficients
• Key insight: Set up coefficient ratios: (coefficient of \(\mathrm{x_1}\))/(coefficient of \(\mathrm{x_2}\)) = (coefficient of \(\mathrm{y_1}\))/(coefficient of \(\mathrm{y_2}\))

3. TRANSLATE the coefficient ratio setup

• From our system: \(\mathrm{\frac{2}{a} = \frac{3}{6}}\)
• Simplifying the right side: \(\mathrm{\frac{3}{6} = \frac{1}{2}}\)
• So our equation becomes: \(\mathrm{\frac{2}{a} = \frac{1}{2}}\)

4. SIMPLIFY to solve for a

• Cross multiply: \(\mathrm{2 \times 2 = a \times 1}\)
• This gives us: \(\mathrm{4 = a}\)
• Therefore: \(\mathrm{a = 4}\)

5. INFER verification (optional but recommended)

• When \(\mathrm{a = 4}\), our system becomes:
  • \(\mathrm{2x + 3y = 12}\)
  • \(\mathrm{4x + 6y = 18}\)
• Dividing the second equation by 2: \(\mathrm{2x + 3y = 9}\)
• Now we have \(\mathrm{2x + 3y = 12}\) and \(\mathrm{2x + 3y = 9}\)
• Same left sides, different right sides (\(\mathrm{12 \neq 9}\)) confirms parallel but distinct lines

Answer: C (4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse "no solution" with "infinite solutions" and incorrectly assume the equations must be identical rather than parallel but distinct.

They might set the entire equations equal: \(\mathrm{2x + 3y = 12}\) should equal \(\mathrm{ax + 6y = 18}\), leading them to try making the equations completely identical. This creates confusion because they can't make both the coefficients AND constants match simultaneously with a single value of a. This leads to confusion and guessing.

Second Most Common Error:

Poor algebraic SIMPLIFY execution: Students set up the correct ratio \(\mathrm{\frac{2}{a} = \frac{3}{6}}\) but make an error when cross-multiplying or solving.

For example, they might incorrectly solve \(\mathrm{\frac{2}{a} = \frac{1}{2}}\) as \(\mathrm{\frac{a}{2} = \frac{1}{2}}\), leading to \(\mathrm{a = 1}\). Or they might flip the fraction incorrectly. This may lead them to select Choice A (2) if they solve \(\mathrm{\frac{a}{2} = 1}\).

The Bottom Line:

This problem tests whether students truly understand what "no solution" means geometrically (parallel lines) and can translate that understanding into algebraic coefficient relationships. The key insight is that parallel lines maintain proportional coefficients but remain distinct equations.