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
$\frac{27}{x-2}+\frac{31}{y+3}=85$
$\frac{31}{x-2}+\frac{27}{y+3}=89$
Let $\frac{1}{x-2}=m \text { and } \frac{1}{y+3}=n$
$27 m+31 n=85 \ldots$
$31 m+27 n=89 \ldots$
Adding both equations
$58 m+58 n=174$
Dividing both sides by $58$
$m + n =3$
Subtracting Eq. $I$ and $II$
$27 m+31 n=85$
$-31 m-27 n=-89$
$-4 m+4 n=-4$
Dividing both sides by $4$
$- m + n =-1 \ldots( IV )$
Equating Eq. $III$ and $IV$
$m + n =3$
$\frac{- m + n =-1}{2 n =2}$
$n =\frac{2}{2}$
$n =1$
Subsituting $n=1$ in Eq. $III$
$m +1=3$
$m=3-1$
$m=2$
$\therefore m =\frac{1}{ x -2} \Rightarrow \frac{1}{ x -2}=2 \Rightarrow 2( x -2)=1 \Rightarrow 2 x -4=1 \Rightarrow 2 x =4+1$
$\Rightarrow 2 x =5 \Rightarrow x =\frac{5}{2}$
$\therefore n =\frac{1}{ y +3} \Rightarrow \frac{1}{ y +3}=1 \Rightarrow y +3=1 \Rightarrow y =1-3 \Rightarrow y =-2$
$y =2$
Hence $(x, y)=\left(\frac{5}{2},-2\right)$
$\frac{31}{x-2}+\frac{27}{y+3}=89$
Let $\frac{1}{x-2}=m \text { and } \frac{1}{y+3}=n$
$27 m+31 n=85 \ldots$
$31 m+27 n=89 \ldots$
Adding both equations
$58 m+58 n=174$
Dividing both sides by $58$
$m + n =3$
Subtracting Eq. $I$ and $II$
$27 m+31 n=85$
$-31 m-27 n=-89$
$-4 m+4 n=-4$
Dividing both sides by $4$
$- m + n =-1 \ldots( IV )$
Equating Eq. $III$ and $IV$
$m + n =3$
$\frac{- m + n =-1}{2 n =2}$
$n =\frac{2}{2}$
$n =1$
Subsituting $n=1$ in Eq. $III$
$m +1=3$
$m=3-1$
$m=2$
$\therefore m =\frac{1}{ x -2} \Rightarrow \frac{1}{ x -2}=2 \Rightarrow 2( x -2)=1 \Rightarrow 2 x -4=1 \Rightarrow 2 x =4+1$
$\Rightarrow 2 x =5 \Rightarrow x =\frac{5}{2}$
$\therefore n =\frac{1}{ y +3} \Rightarrow \frac{1}{ y +3}=1 \Rightarrow y +3=1 \Rightarrow y =1-3 \Rightarrow y =-2$
$y =2$
Hence $(x, y)=\left(\frac{5}{2},-2\right)$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Find the sum of the first $15$ terms of each of the following sequences having $n^{th}$ term as:
$y_n = 9 - 5n.$
→$y_n = 9 - 5n.$
Find the area of an isoscale triangle each of whose equal sides is 13cm and whose base is 24cm.
If $\sin\text{A}=\frac{9}{41},$ computer cos A and tan A.
→A right circular cylinder having diameter $12\ cm$ and height $15\ cm$ is full ice-cream. The ice-cream is to be filled in cones of height $12\ cm$ and diameter $6\ cm$ having a hemispherical shape on the top. Find the number of such cones which can be filled with ice-cream.
→The circumference of a circle exceeds the diameter by 16.8cm. Find the circumference of the circle.
A building is in the form of a cylinder surmounted by a hemi-spherical vaulted dome and contains $41\Big(\frac{19}{21}\Big)\text{m}^3$ of air. If the internal diameter of dome is equal to its total height above the floor, find the height of the building?
→If the squared difference of the zeroes of the quadratic polynomial $f(x) = x^2 + px + 45$ is equal to $144$, find the value of p.
→The sum of the first n terms of an A.P. is $ 4n^2 + 2n.$ Find the $n^{th}$ term of this A.P.
→In fig. AE is the bisector of the exterior $\angle\text{CAD}$ meeting BC produced in E. If AB = 10cm, AC = 6cm and BC = 12cm, find CE.
→The sum of digits of a two number is 15. The number obtained by reversing the order of digits of the given number exceeds the given number by 9. Find the given number.