X + y = 6; x ^ 2 + y ^ 2 = 36

To solve this system of equations, we can use substitution or elimination method. Let's solve it using substitution method:

From the first equation, we can rewrite it as x = 6 - y.

Now substitute this value of x into the second equation:

(6 - y) ^ 2 + y ^ 2 = 36

Expanding and simplifying:

36 - 12y + y ^ 2 + y ^ 2 = 36

2y ^ 2 - 12y = 0

Divide both sides by 2:

y ^ 2 - 6y = 0

Factor out y:

y(y - 6) = 0

So, either y = 0 or (y - 6) = 0.

If y = 0, substitute this value into the first equation to find x:

x + 0 = 6 x = 6

If (y - 6) = 0, then y = 6. Substitute into the first equation to find x:

x + 6 = 6 x = 0

Therefore, the solutions to the system of equations are: (x, y) = (6, 0) and (0, 6).