Question
Solve the following simultaneous equation $:8v - 3u = 5uv,6v - 5u = -2uv$
✓
Answer
$8 v-3 u=5 u v$
$6 v-5 u=-2 u v $
Dividing both sides of each equation by $uv,$
we get, $\frac{8}{u}-\frac{3}{v}=5 . . . (1)$
$\frac{6}{u}-\frac{5}{v}=-2 . . (2) $
Multiplying $(1)$ by $3$ and $(2)$ by $4,$
we get, $ \frac{24}{u}-\frac{9}{v}=15 \ldots . (3)$
$\frac{24}{u}-\frac{20}{v}=-8 . .(4) $
Subtracting $(4)$ from $(3), $
we get, $\frac{11}{v}=23$
$\Rightarrow v =\frac{11}{23}$
$\therefore \frac{6}{u}-\frac{5}{11} \times 23=-2$
$\Rightarrow \frac{6}{u}-\frac{115}{11}=-2$
$\Rightarrow \frac{6}{u} $
$ =-2+\frac{115}{11}$
$=\frac{-22+115}{11}$
$=\frac{93}{11}$
$\Rightarrow u =\frac{6 \times 11}{93}$
$=\frac{22}{31} $
Thus, the solution set is $\left(\frac{22}{11}, \frac{11}{23}\right)$.
$6 v-5 u=-2 u v $
Dividing both sides of each equation by $uv,$
we get, $\frac{8}{u}-\frac{3}{v}=5 . . . (1)$
$\frac{6}{u}-\frac{5}{v}=-2 . . (2) $
Multiplying $(1)$ by $3$ and $(2)$ by $4,$
we get, $ \frac{24}{u}-\frac{9}{v}=15 \ldots . (3)$
$\frac{24}{u}-\frac{20}{v}=-8 . .(4) $
Subtracting $(4)$ from $(3), $
we get, $\frac{11}{v}=23$
$\Rightarrow v =\frac{11}{23}$
$\therefore \frac{6}{u}-\frac{5}{11} \times 23=-2$
$\Rightarrow \frac{6}{u}-\frac{115}{11}=-2$
$\Rightarrow \frac{6}{u} $
$ =-2+\frac{115}{11}$
$=\frac{-22+115}{11}$
$=\frac{93}{11}$
$\Rightarrow u =\frac{6 \times 11}{93}$
$=\frac{22}{31} $
Thus, the solution set is $\left(\frac{22}{11}, \frac{11}{23}\right)$.
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