Question

The pair of equations $x+2y+5=0$ and $-3x-6y+1=0$ have,

• A. A unique solution
• B. Exactly two solutions
• C. Infinitely many solutions
• D. no solution

Answer 1

The correct option is D

no solution

The explanation for the correct option.

If the equation of two lines are,

$$\begin{gathered} a_{1}x+b_{1}y+c_1=0 \\ {a}_{2}x+b_{2}y+c_2=0 \end{gathered}$$

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

If $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$, the two lines are coincident and have infinite solutions.

If $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$, the two lines are intersecting and have unique solutions.

The pair of equations are$x+2y+5=0$ and $-3x-6y+1=0$ .

Compared with the standard form,

$$\begin{gathered} a_1=1,b_1=2,c_1=5 \\ {a}_2=-3,b_2=-6,c_2=1 \end{gathered}$$

$$\begin{gathered} \frac{a_1}{a_2}=-\frac{1}{3} \\ \frac{b_1}{b_2}=\frac{2}{-6}=-\frac{1}{3} \\ \frac{c_1}{c_2}=\frac{5}{1}=5 \end{gathered}$$

Thus,$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$.

Therefore, the two lines are parallel and have no solutions.

Hence, the option $\left(D\right)$ is correct.