Question

Solve for $u$and $v$, $2\left( {3u - v} \right) = 5uv$; and $2\left( {u + 3v} \right) = 5uv$.

Answer 1

Hint: Here, we are required to find the values of two variables using the given two equations. We would convert these equations in terms of $x$ and $y$ so that the RHS contains only a constant term. Then, we would find the values of $x$ and $y$ by eliminating the equations and substituting their values would give us the required values of $u$ and $v$.

Complete step-by-step answer:
Given two equations are:
$2\left( {3u - v} \right) = 5uv$
$ \Rightarrow 6u - 2v = 5uv$
Dividing both sides by$uv$, we get,
$ \Rightarrow \dfrac{{6u}}{{uv}} - \dfrac{{2v}}{{uv}} = \dfrac{{5uv}}{{uv}}$
$ \Rightarrow \dfrac{6}{v} - \dfrac{2}{u} = 5$…………………………(1)
Similarly, second equation can be written as:
$2\left( {u + 3v} \right) = 5uv$
$ \Rightarrow 2u + 6v = 5uv$
Dividing both sides by$uv$, we get,
$ \Rightarrow \dfrac{{2u}}{{uv}} + \dfrac{{6v}}{{uv}} = \dfrac{{5uv}}{{uv}}$
$ \Rightarrow \dfrac{2}{v} + \dfrac{6}{u} = 5$…………………………(2)
Now,
Let $\dfrac{1}{v} = x$ and $\dfrac{1}{u} = y$
Hence, equation (1) becomes:
$6x - 2y = 5$
And, equation (2) becomes:
$2x + 6y = 5$
Now, to solve these equations, we would use elimination method.
To make the coefficients of variable $y$same, we would multiply the first equation by 3.
Hence, the two equations become:
$18x - 6y = 15$
And,
$2x + 6y = 5$
Now, adding both of them, we get,
$\left( {18x - 6y} \right) + \left( {2x + 6y} \right) = 15 + 5$
$ \Rightarrow 20x = 20$
Dividing both sides by 20,
$ \Rightarrow x = 1$
Now, substituting this value in the first equation, i.e.$6x - 2y = 5$
$ \Rightarrow 6\left( 1 \right) - 2y = 5$
$ \Rightarrow 6 - 5 = 2y$
$ \Rightarrow 2y = 1$
Dividing both sides by 2, we get,
$ \Rightarrow y = \dfrac{1}{2}$
Hence, as we know,
$\dfrac{1}{v} = x$ and $x = 1$
$\therefore \dfrac{1}{v} = 1$
Taking reciprocal on both sides,
$ \Rightarrow v = 1$
Similarly,
$\dfrac{1}{u} = y$ and $y = \dfrac{1}{2}$
$\therefore \dfrac{1}{u} = \dfrac{1}{2}$
Taking reciprocal on both sides, we get,
$u = 2$
Hence, the values of $u$ and $v$ are 2 and 1 respectively which satisfies the given equations.

Note: By substituting the values of $u$ and $v$ in the given equations we can check whether our answer is correct or not.
Hence, substituting the values of $u$ and $v$ as 2 and 1 respectively in the first equation,
$2\left( {3u - v} \right) = 5uv$
LHS:
$ = 2\left[ {3\left( 2 \right) - 1} \right]$
$ = 2\left[ {6 - 1} \right]$
$ = 2 \times 5 = 10$
RHS:
$5uv = 5 \times 2{\kern 1pt} \times 1 = 10$
Since,
LHS$ = $RHS$ = $10
Hence, these values satisfy the first equation.
Similarly, substituting these values in the LHS and RHS of the second equation, we would again find them equal. Hence, this shows that our answer is correct and there is not a single step which we have attempted wrong.