$\begin{array}{c}x^{2}+y^{2}=100 \\ x+y=10\end{array}$

Step 1 ：We have a system of two equations: \(x^{2}+y^{2}=100\) and \(x+y=10\).

Step 2 ：The first equation is a circle equation and the second one is a linear equation.

Step 3 ：We can solve this system of equations by substitution or elimination. We will use the substitution method.

Step 4 ：First, we express y from the second equation: \(y = 10 - x\).

Step 5 ：Then, we substitute y in the first equation and solve it for x: \(x^{2} + (10 - x)^{2} - 100 = 0\).

Step 6 ：Solving this equation, we find that the possible values for x are 0 and 10.

Step 7 ：After finding the x values, we can substitute them back into the \(y = 10 - x\) equation to find the corresponding y values.

Step 8 ：When \(x = 0\), \(y = 10\). When \(x = 10\), \(y = 0\).

Step 9 ：Final Answer: The solutions to the system of equations are \(\boxed{(0, 10)}\) and \(\boxed{(10, 0)}\).