Given:
x+ sqrt(y)=7
sqrt(x)+y =11
Solve for integer values of x & y, formally, neither by guessing nor software.
Assume that x is less than y.
Since, (vx, vy)=(11-y, 7-x), it follows that each of vx and vy is an integer.
Now, subtracting the first equation from the second, we obtain:
y-x-(vy-vx)=4
=> (vy-vx)(vy+vx-1)= 4
Since, y>x, it follows that vy> vx
Also, (vy-vx) and (vy+vx-1) must possess different parity.
Then, we must have:
(vy-vx)(vy+vx-1)=1*4, 4*1
=> (vy, vx)=(3,-1), (3,2)
If (vy,vx)=(3,-1)=>(y,x)=(9,1)
Then, x+vy=1+3=4<7. This is a contradiction
If (vy,vx)=(3,2)=> (y,x) =(9,4)
It can easily be verified that (x,y)=(4,9) satisfies each of the given equation s.
Consequently, (x,y)=(4,9) is the only possible solution to the given problem.
Edited on March 26, 2022, 1:14 pm
Puzzle Solution
