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:
Step 1: Add the two equations
Adding the two equations eliminates :
Step 2: Subtract the second equation from the first
Subtracting the second equation from the first eliminates :
Step 3: Solve for and
From :
From :
Step 4: Find all possible solutions
The solutions are all combinations of and :
Final Answer:
The solutions are:
Would you like a detailed explanation of any step or further assistance?
Related Questions:
- How can this method be extended to systems with three variables?
- What are the geometric interpretations of these solutions?
- How would the solution change if the second equation were ?
- Can we use matrix methods to solve this system? If so, how?
- 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
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