Question

find the solution of the pair of linear equations in two variables (22/x+y) + (15/x-y)=5,(55/x+y)+(45/x-y)=14​

Answer 1

$let \: \frac{1}{11} = u \: and \: \frac{1}{x - y} = v$

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-55v=15-14

⟹11u = 1

⟹ u = 1/11

putting u=1/11 in equation (i),we get

$22 \times \frac{1}{11} + 15v = 5$

$⟹2 + 15v = 5$

$⟹15v = 5 - 2$

$⟹15v = 3$

$⟹v = \frac{3}{15}$

$⟹v = \frac{1}{5}$

now,

$u = \frac{1}{x + y}$

$⟹ \frac{1}{x + y} = \frac{1}{11}$

$⟹x + y = 11⠀⠀⠀⠀⠀⠀⠀....(v)$

and,

$v = \frac{1}{x - y}$

$⟹ \frac{1}{x - y} = \frac{1}{5}$

$⟹x - y = 5⠀⠀⠀⠀⠀⠀⠀....(vi)$

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

2x=11+5

⟹2x=16

⟹x=16/2

⟹x=8

putting x= 8 in equation (v),we get

8+y=11

⟹y=11-8

⟹y=3

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