Solving a System of Nonlinear Equations: 2x^2 + y^2 = 20 and x^2

Math Problem Statement

Solve the system of equations: 2x^2 + y^2 = 20 and x^2 - 2y^2 = 22.

We are tasked to solve the following system of equations:

1. & \quad 2x^2 + y^2 = 20 \\ 2. & \quad x^2 - 2y^2 = 22 \end{aligned}$$ Let’s solve this step by step: --- ### Step 1: Solve for one variable From the second equation: $$x^2 = 22 + 2y^2$$ --- ### Step 2: Substitute into the first equation Substitute $$x^2 = 22 + 2y^2$$ into the first equation $$2x^2 + y^2 = 20$$: $$2(22 + 2y^2) + y^2 = 20$$ Simplify: $$44 + 4y^2 + y^2 = 20$$ $$44 + 5y^2 = 20$$ $$5y^2 = -24$$ --- ### Step 3: Analyze Since $$5y^2 = -24$$, it implies $$y^2 < 0$$, which is **not possible** for real numbers. Therefore, the system of equations has **no real solutions**. --- ### Final Answer: The coordinates of the intersection points **do not exist (DNE)**. Would you like more details or have any specific questions? --- ### Related Questions: 1. How do you handle systems of equations when substitution doesn't work directly? 2. What happens when solutions exist in complex numbers instead of real numbers? 3. Can these types of systems represent geometric objects like ellipses or hyperbolas? 4. How can you check if two curves intersect without solving equations? 5. What are some efficient methods for solving nonlinear systems numerically? ### Tip: Always analyze the discriminant or feasibility of your equations early to save effort solving impossible cases!

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Algebra
Systems of Equations
Quadratic Equations

Substitution method
Solving quadratic equations

No real solution theorem (non-negative value of squared terms)

Grades 10-12

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