Question

If √x+y=11,√y+x=7.then find the value of x and y..
solve it.

Answer 1

Given

x−−√+y=11(1)(1)x+y=11

x+y√=7(2)(2)x+y=7

Above equation hold true for x,y≥0x,y≥0 substituting eq(2) from eq(1), we get

x−−√+y−x−y√=11−7x+y−x−y=11−7

x−−√−y√−(x−y)=4x−y−(x−y)=4

(x−−√−y√)−(x−−√−y√)(x−−√+y√)=4(x−y)−(x−y)(x+y)=4

(x−−√−y√)(1−x−−√−y√)=4(x−y)(1−x−y)=4

Case 1: (x−−√−y√)(1−x−−√−y√)=4⋅1(x−y)(1−x−y)=4⋅1

x−−√−y√=4, 1−x−−√−y√=1⟹(x,y)=(4,9) or (9,4)x−y=4, 1−x−y=1⟹(x,y)=(4,9) or (9,4)

Case 2: (x−−√−y√)(1−x−−√−y√)=1⋅4(x−y)(1−x−y)=1⋅4

x−−√−y√=1, 1−x−−√−y√=4⟹(x,y)=(4,9) or (9,4)x−y=1, 1−x−y=4⟹(x,y)=(4,9) or (9,4)

Case 3: (x−−√−y√)(1−x−−√−y√)=2⋅2(x−y)(1−x−y)=2⋅2

x−−√−y√=2, 1−x−−√−y√=2⟹x=25/4,y=9/4x−y=2, 1−x−y=2⟹x=25/4,y=9/4

case-3 does not satisfy original equations hence the solution is x=4,y=9

hope it help you