Mathematics — Standard (041) — CBSE Class 10 board question

Rewrite: $x + 2y + 5 = 0$ and $-3x - 6y + 1 = 0$.

Here, $\dfrac{a_1}{a_2} = \dfrac{1}{-3}$, $\dfrac{b_1}{b_2} = \dfrac{2}{-6} = \dfrac{1}{-3}$, $\dfrac{c_1}{c_2} = \dfrac{5}{1} = 5$.

Since $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$, the pair has no solution. → Option D

• Rearrange both equations to standard form $ax + by + c = 0$ before comparing ratios.
• $-3x = 6y - 1$ becomes $-3x - 6y + 1 = 0$.
• The condition $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$ means parallel lines → no solution (inconsistent pair). This is the key criterion from Section 3.2/Table 3.1.