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

$3 x-5 y=20$

 

$6 x-10 y=40$

Solution:

GIVEN: 

$3 x-5 y=20$

$6 x-10 y=40$

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,

$\frac{3}{6}=\frac{-5}{-10}=\frac{20}{40}$

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

Since $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$ which means $\frac{1}{2}=\frac{1}{2}=\frac{1}{2}$ hence the system of equation has infinitely many solution.

Hence the system of equation has infinitely many solutions

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