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In the following system of equations determine whether the system has a unique solution, no solution or infinite solutions. In case there is a unique solution, find it:
$x\ −\ 3y\ =\ 3$
$3x\ −\ 9y\ =\ 2$
 Given: The given system of equation is $x\ −\ 3y\ =\ 3$ ; $3x\ −\ 9y\ =\ 2$
To do: Solve the system of equation and determine if the solution has unique , infinite or no solution.
Solution:
The given system of equations can be written as:
$x\ −\ 3y\ =\ 3$
$3x\ −\ 9y\ =\ 2$
The given system of equation is of form:
$a_{1} x+b_{1} y=c_{1}\ and\ a_{2} x+b_{2} y=c_{2}$
Here, $a_1 = 1, b_1=-3, c_1=3 \ and \ a_2=3, b_2=-9, c_2=2 $
We have,
$\frac{a_1}{a_2}=\frac{1}{3}$
$\frac{b_1}{b_2}=\frac{-3}{-9}=\frac{1}{3}$
$\frac{c_1}{c_2}=\frac{3}{2} $
Clearly, $\frac{a_1}{a_2}=\frac{b_1}{b_2}$ not equal to $\frac{c_1}{c_2}$
So, given system of equation has no solution
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