Question

Solve the following pair of equations:
$8v-3u=5uv$
$6v-5u=-2uv$

• A. $u=0,v=0$
• B. $u=\frac{22}{31},v=\frac{11}{23}$
• C. Both A and B
• D. None of these

Answer 1

The correct option is B

$u=\frac{22}{31},v=\frac{11}{23}$

Divide the given equations by $uv$,

$\Rightarrow 8v-3u=5uv$
$\Rightarrow \frac{8}{u}-\frac{3}{v}=5...\left(1\right)$

$\Rightarrow 6v-5u=-2uv$
$\Rightarrow \frac{6}{u}-\frac{5}{v}=-2...\left(2\right)$

Assume $\frac{1}{u}=x$ and $\frac{1}{v}=y$

Put the values of $\frac{1}{u}$ and $\frac{1}{v}$ in (1) and (2)

$8x-3y=5...\left(3\right)$

$6x-5y=-2...\left(4\right)$

Multiplying equation (3) by 5 and (4) by 3, we get

$40x-15y=25...\left(5\right)$

$18x-15y=-6...\left(6\right)$

Subtracting (6) from (5), we get

$40x-15y-18x+15y=25+6$
$\Rightarrow 22x=31$
$\Rightarrow x=\frac{31}{22}$

Plugging this value in equation (3), we obtain

$y=\frac{23}{11}$

So, $u=\frac{22}{31}$ and $v=\frac{11}{23}$