https://file.aiquickdraw.com/mathbot/file/1725664821985ea3zer90.jpg

The system of equations you provided is:

\[
\begin{cases}
x^2 + y^2 = 7 \\
x^2 - y^2 = 1
\end{cases}
\]

Let's solve this system step by step.

### Step 1: Add the equations

We add both equations to eliminate \( y^2 \):

\[
(x^2 + y^2) + (x^2 - y^2) = 7 + 1
\]

This simplifies to:

\[
2x^2 = 8
\]

Solving for \( x^2 \):

\[
x^2 = 4
\]

Thus,

\[
x = \pm 2
\]

### Step 2: Solve for \( y^2 \)

Now substitute \( x^2 = 4 \) into one of the original equations. Let's use the first equation \( x^2 + y^2 = 7 \):

\[
4 + y^2 = 7
\]

Solving for \( y^2 \):

\[
y^2 = 7 - 4 = 3
\]

Thus,

\[
y = \pm \sqrt{3}
\]

### Final Solution:

The solutions to the system of equations are the pairs:

\[
(x, y) = (2, \sqrt{3}), (2, -\sqrt{3}), (-2, \sqrt{3}), (-2, -\sqrt{3})
\]

Would you like further details or have any questions?

Here are 5 related questions to expand on this:

1. How would the solution change if one of the constants in the system was modified?
2. Can you solve this system geometrically by interpreting the curves?
3. What would happen if you subtract the two original equations instead of adding them?
4. How would you solve this system using substitution rather than elimination?
5. Can you find the discriminants or test for special cases when dealing with systems of quadratic equations?

**Tip:** For systems of equations involving squares, adding and subtracting equations is a powerful method to simplify the system by eliminating terms.

Solve the system of equations: x^2 + y^2 = 7 and x^2 - y^2 = 1

Learn how to solve a system of quadratic equations where x^2 + y^2 = 7 and x^2 - y^2 = 1. This step-by-step solution walks through elimination and substitution methods to find the values of x and y. Suitable for high school students in Grades 9-11.

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