Here are the essential concepts you must grasp in order to answer the question correctly.
Systems of Equations
A system of equations consists of two or more equations that share common variables. The goal is to find the values of these variables that satisfy all equations simultaneously. In this case, we have a nonlinear system involving a quadratic equation and a product of variables, which requires specific methods for solving.
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Quadratic Equations
A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. In the given system, the equation 2x^2 + y^2 = 18 represents a conic section, specifically an ellipse. Understanding the properties of quadratic equations is essential for manipulating and solving them within a system.
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Substitution and Elimination Methods
Substitution and elimination are two common methods for solving systems of equations. The substitution method involves solving one equation for a variable and substituting that expression into the other equation. The elimination method involves adding or subtracting equations to eliminate a variable. Choosing the appropriate method can simplify the process of finding solutions in nonlinear systems.
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