27/(x-2 )+31/(y+3 )=85; 31/(x-2)+27/(y+3)=89.

iv. 27/(x-2 )+31/(y+3 )=85;    31/(x-2)+27/(y+3)=89.

Solution:

 Let 1/(x-2)=a  and 1/(y+3)=b

∴ the given equations become

27a+31b=85 .................... eq. no. 1

31a+27b=89 .................... eq. no. 2

Adding equation (1) and (2)

27a+13 b=85

31a+27 b=89

58a+58b=174

∴ 58(a+b)= 174

∴ a+b=174/58

∴ a+b=3 ................. eq. no. 3

Now, subtracting equation (1) and (2)

27a+13 b=85

31a+27 b=89

(-)   (-)    (-)

-4a + 4b = -4

∴ - 4 (a – b) = - 4

∴ a – b = -4/-4

∴ a – b = 1 .......... eq. no. 4

Adding equation (3) and (4)

a + b = 3

a – b = 1

2a     = 4

∴ a = 4/2

∴ a = 2

Substituting a = 2  in equation (3)

a + b = 3

∴ 2 + b = 3

∴ b = 3 – 2

∴ b = 1

Now, Re substituting

a = 1/(x – 2)  and b = 1/( y + 3)

a = 2 ∴ 1/(x – 2) = 2 ∴ 1 = 2x – 4 ∴ 1 + 4 = 2x ∴ 5 = 2x ∴ x = 5/2

b = 1 ∴ 1/(y + 3 ) = 1 ∴ 1 = y + 3 ∴ 1 – 3 = y ∴ y = - 2

Ans. x = ⁵/₂  and y = - 2