Question
Solve the following simultaneous equation.
$\frac{27}{x-2}+\frac{31}{y+3}=85 ; \frac{31}{x-2}+\frac{27}{y+3}=89$
$\frac{27}{x-2}+\frac{31}{y+3}=85 ; \frac{31}{x-2}+\frac{27}{y+3}=89$
✓
Answer
$\begin{aligned}& \frac{27}{x-2}+\frac{31}{y+3}=85 \\
& \frac{31}{x-2}+\frac{27}{y+3}=89
\end{aligned}$
Let $\frac{1}{x-2}=m$ and $\frac{1}{y+3}=n$
$27 m+31 n=85 \dots (1)$
$31 m+27 n=89 \dots (2)$
Adding both equations
$58 m+58 n=174$
Dividing both sides by 58
$m + n =3 \dots (3)$
Subtracting Eq. I and II
$\begin{aligned}
& 27 m+31 n=85 \\
& -31 m-27 n=-89 \\
& -4 m+4 n=-4
\end{aligned}$
Dividing both sides by 4
$-m+n=-1 \dots (4)$
Equating Eq. III and IV
$\begin{gathered}
m+n=3 \\
-m+n=-1 \\
\hline 2 n=2
\end{gathered}$
$\begin{aligned}
& n =\frac{2}{2} \\
& n =1
\end{aligned}$
Subsituting $n=1$ in Eq. III
$\begin{aligned}
& m+1=3 \\
& m=3-1 \\
& m=2
\end{aligned}$
$\begin{aligned}
& \therefore m =\frac{1}{ x -2} \\
& \Rightarrow \frac{1}{ x -2}=2 \Rightarrow 2( x -2)=1 \\
& \Rightarrow 2 x -4=1 \Rightarrow 2 x =4+1 \\
& \Rightarrow 2 x =5 \Rightarrow x =\frac{5}{2} \\
& \therefore n =\frac{1}{ y +3} \\
& \Rightarrow \frac{1}{ y +3}=1 \\
& \Rightarrow y +3=1 \\
& \Rightarrow y =1-3 \\
& \Rightarrow y =-2 \\
& y =2
\end{aligned}$
Hence $(x, y)=\left(\frac{5}{2},-2\right)$
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