To solve this system of nonlinear equations, we start by writing out the equations clearly:

1. $x^2 + y^2 = 181$
2. $x^2 - y^2 = -19$

We can solve these equations by adding and subtracting them, which will help us eliminate one of the variables.

Step 1: Add the Equations

Adding both equations:

$(x^2 + y^2) + (x^2 - y^2) = 181 + (-19)$

This simplifies to:

$2x^2 = 162$

Solving for $x^2$:

$x^2 = \frac{162}{2} = 81$

Then,

$x = \pm \sqrt{81} = \pm 9$

Step 2: Substitute $x$ back to Find $y$

Now that we have $x = \pm 9$, we substitute each value into the first equation to find $y$.

Case 1: $x = 9$

Substitute $x = 9$ into $x^2 + y^2 = 181$:

$9^2 + y^2 = 181$

$81 + y^2 = 181$

$y^2 = 181 - 81 = 100$

So,

$y = \pm \sqrt{100} = \pm 10$

Thus, when $x = 9$, we have $y = 10$ or $y = -10$, giving solutions $(9, 10)$ and $(9, -10)$.

Case 2: $x = -9$

Substitute $x = -9$ into $x^2 + y^2 = 181$:

$(-9)^2 + y^2 = 181$

$81 + y^2 = 181$

$y^2 = 100$

So,

$y = \pm \sqrt{100} = \pm 10$

Thus, when $x = -9$, we have $y = 10$ or $y = -10$, giving solutions $(-9, 10)$ and $(-9, -10)$.

Final Answer

The real solutions to the system are:

$(9, 10), (9, -10), (-9, 10), (-9, -10)$

Answer Choices

Select the option that includes these points. If there's a fill-in option, input these four points.

Would you like further clarification on any step, or do you have additional questions?

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Here are some related questions you might explore:

1. How would the solution change if one of the equations had a different constant?
2. Can this method be used to solve all systems of nonlinear equations?
3. What if we only want positive values of $x$ and $y$?
4. How could we verify these solutions graphically?
5. Could there be complex solutions to this system?

Tip: When solving nonlinear equations, combining equations by addition or subtraction can often help simplify or eliminate variables.