Question:
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
Solution:
(d) Given, equations are $x+2 y+5=0$ and $-3 x-6 y+1=0$
Here, $\quad a_{1}=1, b_{1}=2, c_{1}=5$ and $a_{2}=-3, b_{2}=-6, c_{2}=1$
$\therefore \quad \frac{a_{1}}{a_{2}}=-\frac{1}{3}, \frac{b_{1}}{b_{2}}=-\frac{2}{6}=-\frac{1}{3}$,
$\frac{c_{1}}{c_{2}}=\frac{5}{1}$
$\therefore$ $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
Hence, the pair of equations has no solution.