Question

The solution of a pair of linear equations $x + 2y + 5 = 0$ and $-3x - 6y + 1 = 0$ will be:

• A. Unique
• B. Two
• C. Infinitely many
• D. None of the above

Hint:

If $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, the lines are parallel and have no solution.

Answer 1

The Correct Option is D

Step 1: Identify coefficients.
For the equations: 1) $x + 2y + 5 = 0$, coefficients are $a_1 = 1$, $b_1 = 2$, $c_1 = 5$.
2) $-3x - 6y + 1 = 0$, coefficients are $a_2 = -3$, $b_2 = -6$, $c_2 = 1$.

Step 2: Check the ratios.
\[ \frac{a_1}{a_2} = \frac{1}{-3}, \quad \frac{b_1}{b_2} = \frac{2}{-6} = \frac{1}{-3}, \quad \frac{c_1}{c_2} = \frac{5}{1} = 5 \] Since $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, the lines are parallel and have no solution.

Step 3: Conclusion.
Therefore, the pair of linear equations has no solution.