Find all solutions of the system of equations. (If there is no solution, enter NO SOLUTION. Order your answers from smallest to largest x, then from smallest to largest y.)
 

x2	 + 	y2	 = 	24
x2	 − 	y2	 = 	6

We are tasked with solving the system of equations:

1. \( x^2 + y^2 = 24 \)
2. \( x^2 - y^2 = 6 \)

### Step-by-Step Solution:

#### Step 1: Add and subtract the equations.
Let’s add and subtract the two equations to simplify.

**Adding the equations**:
\[
(x^2 + y^2) + (x^2 - y^2) = 24 + 6
\]
\[
2x^2 = 30 \quad \implies \quad x^2 = 15
\]

**Subtracting the equations**:
\[
(x^2 + y^2) - (x^2 - y^2) = 24 - 6
\]
\[
2y^2 = 18 \quad \implies \quad y^2 = 9
\]

#### Step 2: Solve for \( x \) and \( y \).
From \( x^2 = 15 \), we have:
\[
x = \pm \sqrt{15}
\]

From \( y^2 = 9 \), we have:
\[
y = \pm 3
\]

#### Step 3: Determine all combinations of \( x \) and \( y \).
The possible solutions are all combinations of \( x = \pm \sqrt{15} \) and \( y = \pm 3 \):
1. \( (\sqrt{15}, 3) \)
2. \( (\sqrt{15}, -3) \)
3. \( (-\sqrt{15}, 3) \)
4. \( (-\sqrt{15}, -3) \)

### Final Answer:
\[
(\sqrt{15}, 3), (\sqrt{15}, -3), (-\sqrt{15}, 3), (-\sqrt{15}, -3)
\]

Let me know if you'd like further clarification or have questions! 😊

#### Related Questions:
1. How do we verify solutions for this system of equations?
2. What happens if we have \( x^2 + y^2 = 24 \) and \( x^2 - y^2 = -6 \)?
3. Can we graph these equations to visualize the solutions?
4. How does the symmetry of the system influence the solutions?
5. What are some real-world applications of systems of equations like this?

#### Tip:
When solving symmetric systems of equations, look for ways to simplify by adding or subtracting equations!

Find all solutions of the system of equations. (If there is no solution, enter NO SOLUTION. Order your answers from smallest to largest x, then from smallest to largest y.)

x^2 + y^2 = 24
x^2 - y^2 = 6

Discover how to solve the system of equations x^2 + y^2 = 24 and x^2 - y^2 = 6. This detailed explanation demonstrates algebraic methods such as addition and subtraction of equations to find all solutions. Suitable for high school students studying systems of equations.

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