The pair of equations $x + 2y + 5 = 0$ and $-3x - 6y + 1 = 0$ has:
(a) A unique solution
(b) Exactly two solutions
(c) Infinitely many solutions
(d) No solution
Step-by-Step Solution
Key Concept: Compare ratios $\dfrac{a_1}{a_2}, \dfrac{b_1}{b_2}, \dfrac{c_1}{c_2}$. If $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} <br>eq \dfrac{c_1}{c_2}$, the lines are parallel and have no solution.
Here $a_1=1, b_1=2, c_1=5$ and $a_2=-3, b_2=-6, c_2=1$. [0.5 Mark]
$\dfrac{a_1}{a_2} = \dfrac{1}{-3} = -\dfrac{1}{3}$, $\dfrac{b_1}{b_2} = \dfrac{2}{-6} = -\dfrac{1}{3}$, $\dfrac{c_1}{c_2} = \dfrac{5}{1} = 5$.
Since $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2}
eq \dfrac{c_1}{c_2}$, the system has no solution. [0.5 Mark]
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🎯 Official CBSE Marking Scheme:
Finding ratio of coefficients: 0.5 Mark
Comparing ratios and concluding no solution: 0.5 Mark
Correct Answer: No solution