Solve the given equation $2\left( {3u - v} \right) = 5uv$; $2\left( {u + 3v} \right) = 5uv$

Hint: Here we need to solve the given two equations i.e. we need to find the values of the two variables used in these two equations. We will use the elimination method to solve this problem. We will first simplify both the equation and then one from the other to get one variable in terms of the other. Then we will substitute the obtained variable in one of the equations to get the value of another variable. Simplifying it, we will get both the variables.

Complete step-by-step answer:
Here we need to solve the given two equations i.e. we need to find the values of the two variable used in these two equations.
Here we will use the elimination method to solve this problem.
The first equation is:
$2\left( {3u - v} \right) = 5uv$
On multiplying the terms, we get
$ \Rightarrow 6u - 2v = 5uv$………….. $\left( 1 \right)$
The second equation is:
$2\left( {u + 3v} \right) = 5uv$
On multiplying the terms, we get
 $ \Rightarrow 2u + 6v = 5uv$…………… $\left( 2 \right)$
On subtracting equation $\left( 2 \right)$ from equation $\left( 1 \right)$.
$ \underline \
  6u - 2v = 5uv \\
   \pm 2u \pm 6v = \pm 5uv \\
    \\
  4u - 8v = 0 \\ $
So we can write it as
$ \Rightarrow 4u = 8v$
Now, we will divide both sides by 4, we get
$ \Rightarrow \dfrac{{4u}}{4} = \dfrac{{8v}}{4}$
On further simplification, we get
$ \Rightarrow u = 2v$ …………. $\left( 3 \right)$
Now, we will substitute the value of $u$ in equation $\left( 1 \right)$. Therefore, we get
$6 \times 2v - 2v = 5 \times 2v \times v$
On multiplying the terms, we get
$ \Rightarrow 12v - 2v = 10{v^2}$
On subtracting the terms, we get
$ \Rightarrow 10v = 10{v^2}$
On further simplification, we get
$ \Rightarrow 10v = 10v \times v$
Dividing both side by $10v$, we get
$ \Rightarrow v = 1$ ………………….. $\left( 4 \right)$
Now, we will substitute the value of $v$ in equation $\left( 1 \right)$.
$6u - 2 \times 1 = 5 \times u \times 1$
On multiplying the terms, we get
$ \Rightarrow 6u - 2 = 5u$
On subtracting the like terms, we get
$ \Rightarrow 6u - 5u = 2$
On further simplification, we get
$ \Rightarrow u = 2$ ……………….. $\left( 5 \right)$
Hence, the required values of the variables are:-
$ v = 1 \\
  u = 2 \\ $

Note: Here we have used the elimination method to solve this problem. In elimination method, we will first eliminate one variable from the equation using the second equation and from there, we will get the value of one variable, in the same way, we will find the value of another variable and this is known as the elimination method.