Question

Solve the following systems of equations:
$
  2(3u - v) = 5uv \\
  2(u + 3v) = 5uv \\
 $

Answer 1

Hint – In this question simplify by taking 2 to the right hand side and then equate both the equations as one side of them will be equal, this gives the relation between u and v, so simply substitute this relation back into one of the equations, this will help getting the answer.

Complete step-by-step answer:
Given system of equations
$2\left( {3u - v} \right) = 5uv$................. (1)
$2\left( {u + 3v} \right) = 5uv$................... (2)
Now from equation (1) we have,
$ \Rightarrow \dfrac{5}{2}uv = 3u - v$................... (3)
Now from equation (2) we have,
$ \Rightarrow \dfrac{5}{2}uv = u + 3v$................. (4)
Now as we see in equation (3) and (4) L.H.S are same therefore their R.H.S also be same so equate them we have,
$ \Rightarrow 3u - v = u + 3v$
Now simplify the above equation we have,
$ \Rightarrow 3u - u = 3v + v$
$ \Rightarrow 2u = 4v$
$ \Rightarrow u = 2v$............. (5)
Now substitute this value in equation (3) we have,
$ \Rightarrow \dfrac{5}{2}\left( {2v} \right)v = 3\left( {2v} \right) - v$
$ \Rightarrow 5{v^2} = 5v$
$ \Rightarrow 5v\left( {v - 1} \right) = 0$
$ \Rightarrow 5v = 0{\text{ and }}\left( {v - 1} \right) = 0$
$ \Rightarrow v = 0,1$
Now substitute this value in equation (5) we have,
When v = 0
$ \Rightarrow u = 2\left( 0 \right) = 0$
When v = 1
$ \Rightarrow u = 2\left( 1 \right) = 2$
So the solutions of given system of equation are
$ \Rightarrow \left( {u,v} \right) = \left( {0,0} \right){\text{ and }}\left( {2,1} \right)$
So this is the required solution.

Note – There can be another way to simplify the given equation after taking 2 to the right hand side if we would have subtracted both the equations then simply terms would have cancelled each other, then doing the same will too help getting the answer.