Question

Solve the following system 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. $u=\frac{31}{22},v=\frac{23}{11}$

Answer 1

The correct option is B $u=\frac{22}{31},v=\frac{11}{23}$

Divide the given equations by uv,
$8v-3u=5uv\Rightarrow \frac{8}{u}-\frac{3}{v}=5...\left(1\right)$
$6v-5u=-2uv\Rightarrow \frac{6}{u}-\frac{5}{v}=-2...\left(2\right)$

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)
$8x-3y=5...\left(3\right)$
$6x-5y=-2...\left(4\right)$

Solving equations (3) and (4) we get x and y
Multiply (3) with 5 and (4) with 3 to equate the coefficients of y.
$40x-15y=25$
$18x-15y=-6$

$\Rightarrow x=\frac{31}{22}\Rightarrow u=\frac{22}{31}$

Substituting x in (3)
$8\times \frac{31}{22}-3y=5\to 3y=\frac{69}{11}$

$\to y=\frac{23}{11}\Rightarrow v=\frac{11}{23}$

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