3x + 10 = 2y ax - 5y = 7 In the given system of linear equations, a is a...

1. TRANSLATE the equations to standard form

• Given equations:
  • \(\mathrm{3x + 10 = 2y}\)
  • \(\mathrm{ax - 5y = 7}\)

• Converting first equation to standard form \(\mathrm{Ax + By = C}\):
  \(\mathrm{3x + 10 = 2y}\) → \(\mathrm{3x - 2y = -10}\)

• Second equation is already in standard form: \(\mathrm{ax - 5y = 7}\)

2. INFER the condition for no solution

• A system has no solution when the lines are parallel but not identical
• This happens when coefficient ratios are equal but constant ratios are different
• Condition: \(\mathrm{A_1/A_2 = B_1/B_2 \neq C_1/C_2}\)

3. SIMPLIFY by setting up the coefficient ratio equation

• From our equations: \(\mathrm{3x - 2y = -10}\) and \(\mathrm{ax - 5y = 7}\)
• Coefficient ratios must be equal: \(\mathrm{3/a = (-2)/(-5)}\)
• This gives us: \(\mathrm{3/a = 2/5}\)

4. SIMPLIFY to solve for a

• Cross-multiply: \(\mathrm{3 \times 5 = 2a}\)
• \(\mathrm{15 = 2a}\)
• \(\mathrm{a = 7.5}\)

5. Verify the no-solution condition

• With \(\mathrm{a = 7.5}\): coefficient ratio = \(\mathrm{3/7.5 = 2/5 = 0.4}\)
• Constant ratio = \(\mathrm{(-10)/7 \approx -1.43}\)
• Since \(\mathrm{0.4 \neq -1.43}\), we confirm no solution exists

Answer: D) 7.5

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. Instead, they might try to solve the system directly by substitution or elimination, leading to confusion when they get contradictory results like "0 = 15" or similar.

This leads to confusion and guessing rather than systematic analysis.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up \(\mathrm{3/a = 2/5}\) but make arithmetic errors in cross-multiplication. They might calculate \(\mathrm{3 \times 5 = 2a}\) as \(\mathrm{15 = 2a}\) but then incorrectly solve as \(\mathrm{a = 15/2 = 7.5}\), or worse, forget to divide and think \(\mathrm{a = 15}\).

This may lead them to select Choice E (15).

The Bottom Line:

This problem requires students to shift from computational thinking (solving systems) to analytical thinking (understanding when systems can't be solved). The key insight is recognizing that "no solution" is a geometric concept about parallel lines that translates to an algebraic relationship between coefficients.