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You have a system of two equations:

\[
\begin{cases}
x^2 - y^2 = 7 \\
x^2 + y^2 = 25
\end{cases}
\]

To find the values of \(x\) and \(y\), we can solve this system algebraically by using a combination of addition or subtraction. 

### Step 1: Add the two equations

If we add the two equations, the \(y^2\) terms cancel out:

\[
(x^2 - y^2) + (x^2 + y^2) = 7 + 25
\]
\[
2x^2 = 32
\]
\[
x^2 = 16
\]
\[
x = \pm 4
\]

### Step 2: Subtract the two equations

Next, subtract the second equation from the first:

\[
(x^2 - y^2) - (x^2 + y^2) = 7 - 25
\]
\[
-2y^2 = -18
\]
\[
y^2 = 9
\]
\[
y = \pm 3
\]

### Step 3: Find the possible pairs

Now, we have two possible values for \(x\) and \(y\):
- \(x = \pm 4\)
- \(y = \pm 3\)

### Step 4: Verify the combinations

We need to check which combinations of \(x\) and \(y\) satisfy both equations. The pairs \((x, y)\) are:
- \(x = 4\), \(y = 3\)
- \(x = 4\), \(y = -3\)
- \(x = -4\), \(y = 3\)
- \(x = -4\), \(y = -3\)

All four pairs satisfy both equations, so the solutions are:

\[
(x, y) = (4, 3), (4, -3), (-4, 3), (-4, -3)
\]

Would you like further details or clarifications?

Here are some related questions to explore:

1. How can we solve this system using substitution instead of addition/subtraction?
2. What geometric shape do these equations represent?
3. Can this system be solved graphically?
4. What happens if we change the constants in the system of equations?
5. Can we solve a similar system where both equations involve products of \(x\) and \(y\)?

**Tip:** When solving systems of equations, using addition or subtraction can simplify finding solutions when terms cancel out.

Find the values of x and y for the system of equations: x^2 - y^2 = 7 and x^2 + y^2 = 25.

Learn how to solve the system of quadratic equations x^2 - y^2 = 7 and x^2 + y^2 = 25 with step-by-step algebraic solutions. This problem covers key algebra concepts and shows how to find multiple valid pairs for x and y. Suitable for students in Grades 9-11.

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