We are given a system of two equations with two variables, $x$ and $y$. The equations are: $x^2 + y^2 = 410$ $x^2 - y^2 = 10$ We need to solve for $x$ and $y$.

1. Problem Description

We are given a system of two equations with two variables, $x$ and $y$. The equations are:

$x^2 + y^2 = 410$

$x^2 - y^2 = 10$

We need to solve for $x$ and $y$.

2. Solution Steps

We have the system of equations:

$x^2 + y^2 = 410$

$x^2 - y^2 = 10$

We can add the two equations to eliminate $y^2$.

$(x^2 + y^2) + (x^2 - y^2) = 410 + 10$

$2x^2 = 420$

$x^2 = \frac{420}{2}$

$x^2 = 210$

So, $x = \pm\sqrt{210}$.

Now, we can substitute $x^2 = 210$ into either of the original equations. Let's use the first equation:

$x^2 + y^2 = 410$

$210 + y^2 = 410$

$y^2 = 410 - 210$

$y^2 = 200$

So, $y = \pm\sqrt{200} = \pm\sqrt{100 \cdot 2} = \pm 10\sqrt{2}$.

Thus, the solutions are $(x,y) = (\sqrt{210}, 10\sqrt{2})$, $(\sqrt{210}, -10\sqrt{2})$, $(-\sqrt{210}, 10\sqrt{2})$, and $(-\sqrt{210}, -10\sqrt{2})$.

3. Final Answer

The solutions are $x = \pm\sqrt{210}$ and $y = \pm 10\sqrt{2}$.

The solutions are:

$(\sqrt{210}, 10\sqrt{2})$

$(\sqrt{210}, -10\sqrt{2})$

$(-\sqrt{210}, 10\sqrt{2})$

$(-\sqrt{210}, -10\sqrt{2})$