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Solve the following systems of equations:
x + y = 2xy
\[\dfrac{{{\text{x - y}}}}{{{\text{xy}}}}{\text{ = 6}}\]
\[{\text{x}} \ne {\text{0,y}} \ne {\text{0}}\]

Last updated date: 15th Jul 2024
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Hint: In order to solve this problem subtract the equations given then solve algebraically to get the values of x and y. Doing this will give you the right answer.

Complete step-by-step answer:
The given equations are:-
x + y = 2xy (1)
\[\dfrac{{{\text{x - y}}}}{{{\text{xy}}}}{\text{ = 6}}\] (2)
\[{\text{x}} \ne {\text{0,y}} \ne {\text{0}}\]
Equation number (2) can be written as:
x – y = 6xy (3)
Adding equation number (1) and (3) we get,
x + y + x - y = 8xy
On cancelling x from both sides of the equation we get the new equation as:
y = $\dfrac{1}{4}$
On putting the value of y in equation (1) we get the new equation as:
x + $\dfrac{1}{4}$= 2x$\left( {\dfrac{1}{4}} \right)$=$\dfrac{{\text{x}}}{2}$
On solving it further we get the equation as:
$\dfrac{{\text{x}}}{{\text{2}}}{\text{ = - }}\dfrac{{\text{1}}}{{\text{4}}}$
Then x = $ - \dfrac{1}{2}$
So, the value of x = $ - \dfrac{1}{2}$ and the value of y = $\dfrac{1}{4}$.

Note: Whenever you face such types of problems you have to simplify the equations and if the number of unknown is equal to the number of equations then you can get the value of performing mathematical operations between two equations. Proceeding like this you will get the right answer.