Question

Solve the following pair of linear equations by the substitution method:

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

Answer 1

$$\begin{gathered} 3x-y-3=0......\left(1\right) \\ 9x-3y-9=0......\left(2\right) \end{gathered}$$

On comparing equation $\left(1\right)$ and $\left(2\right)$, we have

$a_1=3,b_1=-1,c_1=-3$ and $a_2=9,b_2=-3,c_2=-9$

$$\begin{gathered} \frac{3}{9}=\frac{-1}{-3}=\frac{-3}{-9} \\ \Rightarrow \frac{1}{3}=\frac{1}{3}=\frac{1}{3} \\ \Rightarrow \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2} \end{gathered}$$

Therefore, the given equations have infinitely many solutions.