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

Solution:

GIVEN: 

$\begin{aligned} x-2 y &=8 \\ 5 x-10 y &=10 \end{aligned}$

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,

$\frac{1}{5}=\frac{-2}{-10}=\frac{8}{10}$

$\frac{1}{5}=\frac{1}{5} \neq \frac{2}{5}$

Since $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$ which means $\frac{1}{5}=\frac{1}{5} \neq \frac{2}{5}$ hence the system of equation has no solution.