x^1/2 + y = 7
x + y^1/2 = 11
Find the value of x and y
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                                      ----- eq (i)

                              
                                      ----- eq (ii)

We then get: 
         
                

                                 Let 
              Then: 
                         then becomes: 
                        
                        
Using the RATIONAL ROOT THEOREM, we find that a root of the above equation is: t = 2, which makes its
FACTOR, t - 2. When divided by t - 2, using LONG DIVISION of POLYNOMIALS, or using SYNTHETIC DIVISION,
the other factor of , besides t - 2, is: .
From this, we find another REAL solution being approximately 3.13131. The other 2 are negative (< 0) and
so, MUST be REJECTED/IGNORED, since  CANNOT have a negative (< 0) value for t. 

I will continue with the REAL INTEGER value, 2.
      ---- Back-substituting t = 2 for 


        ----- eq (ii)
        x = 11 - 2 ----- Substituting 2 for  in eq (ii)
        x = 9

So, the ONLY INTEGER-solution set is: (x, y) = (9, 4). I'll let you substitute the other REAL VALUE, 3.13131
for t, to determine the other SOLUTION-SET.