Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
$x^2+y^2=02$
$x^2-3y^2=0$

Verified step by step guidance

1. Start by writing down the system of equations clearly: \[x^2 + y^2 = 0\] \[2x^2 - 3y^2 = 0\]
2. From the first equation, observe that the sum of two squares equals zero. Since squares of real numbers are nonnegative, this implies both \(x^2\) and \(y^2\) must be zero for real solutions. However, since complex solutions are allowed, consider the possibility of nonzero complex values.
3. Express one variable in terms of the other using the first equation. For example, from \[x^2 + y^2 = 0\], we get \[y^2 = -x^2\].
4. Substitute \[y^2 = -x^2\] into the second equation \[2x^2 - 3y^2 = 0\] to eliminate \[y^2\] and get an equation in terms of \[x^2\] only: \[2x^2 - 3(-x^2) = 0\].
5. Simplify the resulting equation to solve for \[x^2\], then use \[y^2 = -x^2\] to find \[y^2\]. Finally, take square roots (considering both positive and negative roots) to find all possible values of \[x\] and \[y\], including complex solutions.

Key Concepts

Nonlinear Systems of Equations

A nonlinear system consists of two or more equations where at least one equation is nonlinear, such as involving variables raised to powers other than one. Solving these systems requires finding all variable values that satisfy every equation simultaneously, including real and complex solutions.

Complex Numbers and Imaginary Solutions

When solving equations like x² + y² = 0, solutions may involve complex numbers, which include the imaginary unit i where i² = -1. Understanding how to work with complex numbers allows finding solutions beyond the real number line, especially when sums of squares equal zero.

Substitution and Elimination Methods

These algebraic techniques help solve systems by expressing one variable in terms of another (substitution) or combining equations to eliminate a variable (elimination). Applying these methods simplifies nonlinear systems into solvable forms, enabling the determination of all possible solutions.