If the pair of linear equations : $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ is consistent and dependent, then
Generated by claude-sonnet-4-6 · 2026-06-15 10:34 · grounding rag
Model Answer
Option D: $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$
A consistent and dependent pair has coincident lines (infinitely many solutions), which occurs when all three ratios are equal.
Explanation
Examiners expect you to recall the three conditions directly:
- $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ → consistent (unique solution, intersecting lines)
- $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ → consistent & dependent (infinitely many solutions, coincident lines)
- $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$ → inconsistent (no solution, parallel lines)
"Dependent" always means coincident lines → all three ratios equal → D.