Question

(7u-2v)/uv=5 & (8u+7u)/uv=15

Answer 1

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Answer 2

$\frac{7u - 2v}{uv} = 5 \: \: \: \: \: \: \: \: \: \: \: \: \bf ...(i) \\ \\ \frac{8u + 7u}{uv} = 15 \: \: \: \: \: \: \: \: \: \bf ...(ii)$

$\underline{\underline{\huge\mathfrak{Solution}}}$

$\underline{\mathbf{Equation (i)}}$

$\frac{7u - 2v}{uv} = 5 \\ \\ 7u - 2v = 5uv$

Dividing $\bf uv$ by the whole equation we get :

$\frac{7u}{uv} - \frac{2v}{uv} = \frac{5uv}{uv} \\ \\ \frac{7}{v} - \frac{2}{u} = 5$

$\underline{\mathbf{Equation (ii)}}$

$\frac{8u + 7v}{uv} = 15 \\ \\ 8u + 7v = 5uv$

Dividing $\bf uv$ by the whole equation we get :
$\frac{8u}{uv} + \frac{7v}{uv} = \frac{15uv}{uv} \\ \\ \frac{8}{v} + \frac{7}{u} = 15$

Let us assume $\frac{1}{v}$ be x and $\frac{1}{u}$ be y.
So, we get two equations,

7x - 2y = 5 ...(i)
8x + 7y = 15 ...(ii)

From equation (i)

$7x - 2y = 5 \\ \\ 7x = 5 + 2y \\ \\ \bf x = \frac{5 + 2y}{7}$

Substituting the value of x in Equation (ii) we get,

$8( \frac{5 + 2y}{7} ) + 7y = 15 \\ \\ \frac{40 + 16y}{7} + 7y = 15 \\ \\ \\ \frac{40 + 16y + 49y}{7} = 15 \\ \\ 40 + 65y = 15 \times 7 \\ \\ 40 + 65y = 105 \\ \\ 65y = 105 - 40 \\ \\ 65y = 65 \\ \\ \bf y = 1$

Putting the value of y in equation (i)

7x - 2(1) = 5

7x - 2 = 5

7x = 5 + 2

x = $\frac{7}{7}$

x = 1

We had assumed that $\frac{1}{v}$ = x and $\frac{1}{u}$ = y.

So,

$\frac{1}{v} = 1 \\ \\ v = 1$

and

$\frac{1}{u} = 1 \\ \\ u = 1$

$\mathcal{The \: value \: of \: u \: and \: v \: is \: 1.}$