In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}x^3 + y = 0 \\x^2 - y = 0\end{cases}$

Verified step by step guidance

1. Start with the given system of equations: \(x^3 + y = 0\) and \(x^2 - y = 0\).
2. From the second equation, express \(y\) in terms of \(x\): \(y = x^2\).
3. Substitute \(y = x^2\) into the first equation to eliminate \(y\): \(x^3 + x^2 = 0\).
4. Factor the resulting equation: \(x^2(x + 1) = 0\).
5. Set each factor equal to zero and solve for \(x\): \(x^2 = 0\) or \(x + 1 = 0\), then find corresponding \(y\) values using \(y = x^2\).

Key Concepts

Solving Systems of Equations

A system of equations consists of two or more equations with the same variables. Solving the system means finding all variable values that satisfy every equation simultaneously. Methods include substitution, elimination, and graphing, each useful depending on the system's complexity.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, making it easier to solve. It is especially effective when one equation is already solved for a variable.

Nonlinear Equations and Polynomial Functions

Nonlinear systems include equations with variables raised to powers greater than one, such as x^3 or x^2. These systems can have multiple solutions or complex roots. Understanding polynomial behavior and factoring techniques helps in solving and interpreting these equations.