Question

(x) ^1/2 +y = 11 and x - (y) ^1/2 = 9 solve for x and y.

Answer 1

Solution:
√x + y = 11
X - √y = 9
Let √x be u & √y be v. Therefore,
u + v^2 = 11 .......E1
u^2 - v = 9 .......E2

from E2, we have
v = u^2 - 9 ........E3

substituting the value of v in E1, we get
u + (u^2 - 9)^2 = 11
or u + u^4 - 2xu^2x9 + 81 = 11
here we will have four values of u since it is quartic equation. I've used my calculator (see attached pic) to solve this, since solving this would be a lengthy process. The values are
u = 2.465
u = 3.428
u = - 3.58
u = - 2.313

Now we shall only take the positive cases.
Case 1;
u = 2.465
v = 2.465^2 - 9 .........(from E3)
Therefore, v = 6.076 - 9 = - 2.924
This violates our equations, hence discarded.

Case 2;
u = 3.428
v = 3.428^2 - 9 .........(from E3)
Therefore, v = 11.75 - 9 = 2.75

Check, for values of u & v
3.428 + 2.75^2 = 3.428 + 7.563 = 11 ...(approx 11)
Hence E1 satisfied.

3.428^2 - 2.75 = 11.75 - 2.75 = 9
Hence E2 also satisfied.

Since we have assumed u = √x
Therefore, √x = 3.428
or x = 3.428^2 = 11.75

& v = √y
Therefore, √y = 2.75
or y = 2.75^2 = 7.56

Hence x = 11.75
& y = 7.56