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!