#### Question

Solve the following systems of equations:

`22/(x + y) + 15/(x - y) = 5`

`55/(x + y) + 45/(x - y) = 14`

#### Solution

Let `1/(x + y) = u and 1/(x - y) = v` Then, the given system of equation becomes

22u + 15v = 5 ...(i)

55u + 45v = 14..(ii)

Multiplying equation (i) by 3, and equation (ii) by 1, we get

66u + 45v = 15 ....(iii)

55u + 45v = 14 .....(iv)

Subtracting equation (iv) from equation (iii), we get

66u - 55u = 15 - 4

=> 11u = 1

=> u = 1/11

Putting u = 1/11 in equaiton (i) we get

`22 xx 1/11 + 15v = 5`

=> 2 + 15v = 5

`=> 15v = 5 - 2`

=> 15v = 3

`=> v = 3/5 = 1/5`

Now u = 1/(x + y)

=> 1/(x + y) = 1/11

=> x + y = 11 ....(v)

And v = 1/(x - y)

`=> 1/(x- y) = 1/5`

=> x - y = 5 .....(vi)

Adding equation (v) and equation (vi), we get

2x = 11 + 5

=> 2x = 16

`=> x = 16/2 = 8`

Putting x =- 8 in equation (v), we get

8 + y = 11

y = 11 - 8 = 3

Hence, solution of the given system of equation is x = 8, y = 3