To solve the system of equations, we can use the substitution method.

From the second equation, we have y = 9 - x.

Substitute y = 9 - x into the first equation:

x^2 + (9 - x)^2 = 18
x^2 + 81 - 18x + x^2 = 18
2x^2 - 18x + 81 = 18
2x^2 - 18x + 63 = 0

Now, factor the quadratic equation:

2(x - 3)(x - 7) = 0

Setting each factor to zero, we get:
x - 3 = 0 --> x = 3
x - 7 = 0 --> x = 7

Now, plug the values of x back into y = 9 - x to find the corresponding y values:

For x = 3, y = 9 - 3 = 6
For x = 7, y = 9 - 7 = 2

Therefore, the solutions to the system of equations are (x, y) = (3, 6) and (7, 2).