Question

Without actually solving the simultaneous equations given below,
decide which simultaneous equations have unique solutions ,
no solution or infinitely many solutions.
$\left(\mathrm{ii}\right)3y=2-x;3x=6-9y$

Answer 1

$$\begin{gathered} 3y=2-x \\ 3x=6-9y \end{gathered}$$

These equations can be written as follows:

$$\begin{gathered} x+3y=2 \\ 3x+9y=6 \end{gathered}$$

Comparing the above equations with the equations $a_{1}x+b_{1}y=c_1\mathrm{and}{a}_{2}x+b_{2}y=c_2,\mathrm{we}\mathrm{get}$:

$$\begin{gathered} a_1=1b_1=3c_1=2 \\ {a}_2=3b_2=9c_2=6 \\ \therefore \frac{a_1}{a_2}=\frac{1}{3};\frac{b_1}{b_2}=\frac{3}{9}=\frac{1}{3}\mathrm{and}\frac{c_1}{c_2}=\frac{2}{6}=\frac{1}{3} \end{gathered}$$

$$\begin{gathered} \mathrm{i}.\mathrm{e}.,\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2} \\ \therefore \mathrm{The}\mathrm{given}\mathrm{equations}\mathrm{have}\mathrm{infinitely}\mathrm{many}\mathrm{solutions}. \end{gathered}$$