Solve the the system of equations below. Give your solutions as points with exact coordinates, using radicals, not decimals.

{
x
2
+
y
2
=
9
x
2
−
y
2
=
3
 

The solution(s) are:

We are solving the following system of equations:

1. \( x^2 + y^2 = 9 \)
2. \( x^2 - y^2 = 3 \)

### Step 1: Add the two equations
Adding the two equations eliminates \( y^2 \):
\[
(x^2 + y^2) + (x^2 - y^2) = 9 + 3
\]
\[
2x^2 = 12
\]
\[
x^2 = 6
\]

### Step 2: Subtract the second equation from the first
Subtracting the second equation from the first eliminates \( x^2 \):
\[
(x^2 + y^2) - (x^2 - y^2) = 9 - 3
\]
\[
2y^2 = 6
\]
\[
y^2 = 3
\]

### Step 3: Solve for \( x \) and \( y \)
From \( x^2 = 6 \):
\[
x = \pm\sqrt{6}
\]

From \( y^2 = 3 \):
\[
y = \pm\sqrt{3}
\]

### Step 4: Find all possible solutions
The solutions are all combinations of \( x \) and \( y \):
\[
(x, y) = (\sqrt{6}, \sqrt{3}), (\sqrt{6}, -\sqrt{3}), (-\sqrt{6}, \sqrt{3}), (-\sqrt{6}, -\sqrt{3})
\]

### Final Answer:
The solutions are:
\[
(\sqrt{6}, \sqrt{3}), (\sqrt{6}, -\sqrt{3}), (-\sqrt{6}, \sqrt{3}), (-\sqrt{6}, -\sqrt{3})
\]

Would you like a detailed explanation of any step or further assistance?

---

### Related Questions:
1. How can this method be extended to systems with three variables?
2. What are the geometric interpretations of these solutions?
3. How would the solution change if the second equation were \( x^2 - y^2 = 5 \)?
4. Can we use matrix methods to solve this system? If so, how?
5. How do we confirm the solutions satisfy both equations?

### Tip:
Always check your solutions by substituting them back into the original equations to ensure they satisfy all conditions.

Solve the system of equations below. Give your solutions as points with exact coordinates, using radicals, not decimals.

x^2 + y^2 = 9
x^2 - y^2 = 3

Explore how to solve the system of equations x^2 + y^2 = 9 and x^2 - y^2 = 3 with step-by-step guidance. Discover solutions as exact points using radicals, covering key algebraic concepts and theorems. Suitable for high school students in Grades 9-12.

Math Problem Statement

{ x 2 + y 2

9 x 2 − y 2

Solution

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