Question

Solve
$\frac{33}{u+2}+\frac{12}{v-3}=123$
and $\frac{12}{u+2}+\frac{33}{v-3}=102$

Answer 1

The given simultaneous equations are
$\frac{33}{u+2}+\frac{12}{v-3}=123$ ........ (i)
$\frac{12}{u+2}+\frac{33}{v-3}=102$ ........ (ii)
Substituting $\frac{1}{u+2}=m$ and $\frac{1}{v-3}=n$ then adding, we get
$33m+12n=123$ ....... (iii)
$12m+33n=102$ ....... (iv)
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$45m+45n=225$
$45(m+n)=225$
$m+n=5$ ........ (v)
Substracting equation (iv) from (iii), we get
$33m+12n=123$
$12m+33n=102$
- (-) (-)
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$21m-21n=21$
$21(m-n)=21$
$m-n=1$ ....... (vi)
Now, adding equation (v) and (vi), we get
$m+n=5$
$m-n=1$
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$2m=6$
$m=3$
Substituting m = 3 in equation (v), we get
$m+n=5$
$3+n=5$
$n=2$
Re-substituting values of m and n.
$\frac{1}{u+2}=3$ $\frac{1}{v-3}=2$
$3u+6=1$ $2v-6=1$
$3u=-5$ $2v=1+6$
$2v=7$
$u=\frac{-5}{3}$ $v=\frac{7}{2}$