Question:4/3x - 5/3y - 1/3 = 0qx - 3/5y + 2/5 = 0In the given system of equations, q is...

1. TRANSLATE the problem information

• Given: A system of two linear equations with parameter q
• Find: Value of q that makes the system have no solution

2. INFER what 'no solution' means

• No solution occurs when the lines are parallel (same slope, different y-intercepts)
• Algebraically: coefficient ratios are equal, but constant ratio is different
• This gives us our strategy: set up ratio equations

3. SIMPLIFY by clearing fractions

• First equation × 3: \(4\mathrm{x} - 5\mathrm{y} = 1\)
• Second equation × 5: \(5\mathrm{qx} - 3\mathrm{y} = -2\)
• Now we have cleaner integer coefficients to work with

4. INFER the ratio condition and SIMPLIFY to solve

• For no solution: \(\frac{4}{5\mathrm{q}} = \frac{-5}{-3} = \frac{5}{3}\)
• From \(\frac{4}{5\mathrm{q}} = \frac{5}{3}\), cross multiply: \(4 \cdot 3 = 5\mathrm{q} \cdot 5\)
• SIMPLIFY: \(12 = 25\mathrm{q}\), so \(\mathrm{q} = \frac{12}{25}\)

5. INFER the verification step

• Check that constant ratio differs: \(\frac{1}{-2} = -\frac{1}{2}\)
• Since \(-\frac{1}{2} \neq \frac{5}{3}\), we confirm no solution exists

Answer: B) \(\frac{12}{25}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the conditions for no solution versus infinitely many solutions. They might think that for no solution, ALL ratios (including constants) must be equal, or they might think coefficients must be different.

This conceptual confusion leads them to set up incorrect equations, possibly getting \(\mathrm{q} = -\frac{4}{5}\) or other wrong values, causing them to select an incorrect answer choice or abandon the systematic approach and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when clearing fractions or cross-multiplying, especially with the negative signs and multiple fraction operations involved in this problem.

For example, incorrectly handling \(\frac{-5}{-3}\) as \(\frac{5}{-3}\) instead of \(\frac{5}{3}\), or making errors in cross multiplication. This may lead them to select Choice E) \(-\frac{12}{25}\) (wrong sign) or other incorrect values.

The Bottom Line:

This problem requires understanding the geometric meaning of 'no solution' (parallel lines) and translating that into precise algebraic conditions about coefficient ratios - a connection many students miss.