Math Problem Statement

Solve the system of equations for the intersection point(s) of the two curves: { 2x^2 + y^2 = 17, x^2 - 2y^2 = 34 }. List the coordinates of the intersection point(s). If there are no solutions, enter DNE for Does Not Exist.

Solution

To solve the system of equations for the intersection points:

2x^2 + y^2 = 17 \quad (1) \\ x^2 - 2y^2 = 34 \quad (2) \end{cases}$$ ### Step 1: Solve for $$x^2$$ from Equation (2). From $$x^2 - 2y^2 = 34$$, isolate $$x^2$$: $$x^2 = 2y^2 + 34$$ ### Step 2: Substitute $$x^2$$ into Equation (1). Substitute $$x^2 = 2y^2 + 34$$ into $$2x^2 + y^2 = 17$$: $$2(2y^2 + 34) + y^2 = 17$$ Simplify: $$4y^2 + 68 + y^2 = 17$$ Combine like terms: $$5y^2 + 68 = 17$$ Subtract 68 from both sides: $$5y^2 = -51$$ ### Step 3: Analyze the result. Here, $$5y^2 = -51$$. Since $$y^2$$ cannot be negative in real numbers, there are no real solutions. ### Final Answer: The curves **do not intersect** in real coordinates. Therefore, the solution is: $$\boxed{\text{DNE}}$$ Would you like me to explain further or provide visual insights into why no intersection occurs? --- ### Related Questions: 1. How would the solution change if the second equation were $$x^2 - 2y^2 = 10$$? 2. Can you graph these equations to confirm there’s no intersection? 3. How would you solve this system in the complex number plane? 4. What types of conic sections do these equations represent? 5. How can substitution methods differ from elimination in solving such systems? **Tip**: Always check for negative results when solving for squared terms; they might indicate no real solutions.

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Math Problem Analysis

Mathematical Concepts

Systems of Equations
Substitution Method
Quadratic Equations
Real and Complex Solutions

Formulas

Substitution into equations
Solving quadratic forms: ax^2 + bx + c = 0

Theorems

Properties of Real Numbers (no real solutions for negative squared terms)
System of Equations Solutions

Suitable Grade Level

Grades 10-12

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