We are given a system of two equations: $x^2 + y^2 = 10$ $x^2 - y^2 = 10$ We need to find the values of $x$ and $y$ that satisfy both equations.

1. Problem Description

We are given a system of two equations:

$x^2 + y^2 = 10$

$x^2 - y^2 = 10$

We need to find the values of $x$ and $y$ that satisfy both equations.

2. Solution Steps

We can solve this system of equations by using the elimination method.

Adding the two equations, we have:

$(x^2 + y^2) + (x^2 - y^2) = 10 + 10$

$2x^2 = 20$

$x^2 = \frac{20}{2}$

$x^2 = 10$

$x = \pm \sqrt{10}$

Now we substitute $x^2 = 10$ into the first equation $x^2 + y^2 = 10$:

$10 + y^2 = 10$

$y^2 = 10 - 10$

$y^2 = 0$

$y = 0$

Thus, the solutions are $(\sqrt{10}, 0)$ and $(-\sqrt{10}, 0)$.

3. Final Answer

The solutions are $x = \sqrt{10}$ and $y = 0$, or $x = -\sqrt{10}$ and $y = 0$.

So the solutions are $(\sqrt{10}, 0)$ and $(-\sqrt{10}, 0)$.