Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5. x^2 + y^2 = 0 2x^2 - 3y^2 = 0

Key Concepts

Nonlinear Equations

Nonlinear equations are equations in which the variables are raised to a power greater than one or involve products of variables. Unlike linear equations, which graph as straight lines, nonlinear equations can produce curves, circles, or other complex shapes. Understanding how to manipulate and solve these equations is crucial for finding their solutions.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'i' is the imaginary unit defined as the square root of -1. In the context of solving equations, complex solutions may arise when the discriminant of a quadratic equation is negative, indicating that the solutions cannot be represented on the real number line.

Systems of Equations

A system of equations consists of two or more equations that share common variables. Solving a system involves finding values for the variables that satisfy all equations simultaneously. In nonlinear systems, techniques such as substitution, elimination, or graphical methods may be employed to find all possible solutions, including real and complex ones.