In each of the following systems of equations determine whether the system has a unique solution,
Question:

In each of the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:

$x-3 y=3$

$3 x-9 y=2$

Solution:

GIVEN:

$x-3 y=3$

$3 x-9 y=2$

To find: To determine whether the system has a unique solution, no solution or infinitely many solutions

We know that the system of equations

$a_{1} x+b_{1} y=c_{1}$

$a_{2} x+b_{2} y=c_{2}$

For unique solution

$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$

For no solution

$\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$

For infinitely many solution

$\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$

Here,

$\frac{1}{3}=\frac{3}{9} \neq \frac{3}{2}$

$\frac{1}{3}=\frac{1}{3} \neq \frac{3}{2}$

Since $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ which means $\frac{1}{3}=\frac{1}{3} \neq \frac{3}{2}$ hence the system of equation has no solution.

Hence the system of equation has no solution