Math Problem Statement

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:

Solution

We are solving the following system of equations:

  1. x2+y2=9x^2 + y^2 = 9
  2. x2y2=3x^2 - y^2 = 3

Step 1: Add the two equations

Adding the two equations eliminates y2y^2: (x2+y2)+(x2y2)=9+3(x^2 + y^2) + (x^2 - y^2) = 9 + 3 2x2=122x^2 = 12 x2=6x^2 = 6

Step 2: Subtract the second equation from the first

Subtracting the second equation from the first eliminates x2x^2: (x2+y2)(x2y2)=93(x^2 + y^2) - (x^2 - y^2) = 9 - 3 2y2=62y^2 = 6 y2=3y^2 = 3

Step 3: Solve for xx and yy

From x2=6x^2 = 6: x=±6x = \pm\sqrt{6}

From y2=3y^2 = 3: y=±3y = \pm\sqrt{3}

Step 4: Find all possible solutions

The solutions are all combinations of xx and yy: (x,y)=(6,3),(6,3),(6,3),(6,3)(x, y) = (\sqrt{6}, \sqrt{3}), (\sqrt{6}, -\sqrt{3}), (-\sqrt{6}, \sqrt{3}), (-\sqrt{6}, -\sqrt{3})

Final Answer:

The solutions are: (6,3),(6,3),(6,3),(6,3)(\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 x2y2=5x^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.

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Equations
Radicals

Formulas

Sum and difference of squares
x^2 + y^2 = r^2 (circle equation)

Theorems

Addition and Subtraction Method for Solving Systems of Equations

Suitable Grade Level

Grades 9-12