Find the ordered pair of real numbers (x,y) that satsfies x + y + sqrt(x + y) = 12 and x - y + sqrt(x - y) = 6.
+128475 x + y + sqrt (x +y) = 12
x - y + sqrt (x - y ) = 6
Let ( x +y) = a
Let (x - y) = b
So we have
a + sqrt (a) =12
a + sqrt (a) - 12 = 0 Factor as
(sqrt (a) + 4) ( sqrt (a) - 3) = 0
sqrt (a) + 4 = 0 sqrt (a) - 3 = 0
No real solutions exist sqrt (a) = 3
a = 9
And
b + sqrt (b) = 6
b +sqrt (b) - 6 = 0 Factor as
(sqrt (b) + 3) (sqrt (b) - 2) = 0
sqrt (b) + 3 = 0 sqrt (b) - 2 = 0
No real solutions exist sqrt (b) = 2
b = 4
So
x + y = 9
x - y = 4 add these
2x = 13
x = 13/2
And
13/2 + y =9
13/2 + y = 18/2
y = 5/2
(x, y) = ( 13/2, 5/2)
