To solve the system of equations given by:

\[
\begin{aligned}
1. & \quad X^2 + Y^2 = 1 \\
2. & \quad X^2 - Y^2 = 1 
\end{aligned}
\]

We can approach this by solving the system step-by-step.

### Step 1: Add the two equations
Adding the two equations will help us eliminate \( Y^2 \):

\[
(X^2 + Y^2) + (X^2 - Y^2) = 1 + 1
\]

This simplifies to:

\[
2X^2 = 2
\]

Dividing both sides by 2:

\[
X^2 = 1
\]

Taking the square root of both sides:

\[
X = \pm 1
\]

### Step 2: Substitute \( X \) back into one of the original equations
Let's substitute \( X = 1 \) and \( X = -1 \) into one of the original equations. We'll use the first equation:

\[
X^2 + Y^2 = 1
\]

**Case 1: \( X = 1 \)**

\[
1^2 + Y^2 = 1 \implies 1 + Y^2 = 1 \implies Y^2 = 0
\]

\[
Y = 0
\]

**Case 2: \( X = -1 \)**

\[
(-1)^2 + Y^2 = 1 \implies 1 + Y^2 = 1 \implies Y^2 = 0
\]

\[
Y = 0
\]

### Step 3: Write the solutions
The solutions to the system of equations are:

\[
(X, Y) = (1, 0) \quad \text{or} \quad (X, Y) = (-1, 0)
\]

### Summary
The system has two solutions: \((1, 0)\) and \((-1, 0)\).

Would you like more details on any of the steps? Here are some related questions you might find interesting:

1. How do we solve a system of non-linear equations in general?
2. What are the graphical interpretations of these solutions?
3. How do we approach systems where the variables aren't squared?
4. What happens if one equation is quadratic and the other linear?
5. How would the solutions change if one of the equations were modified slightly?

**Tip:** Always double-check your solutions by substituting them back into the original equations to ensure they satisfy both equations.

Explore how to solve the system of equations X² + Y² = 1 and X² - Y² = 1 using algebraic methods. This problem highlights key concepts in solving systems of quadratic equations and is suitable for students in Grades 9-11.

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