In Exercises 29–42, solve each system by the method of your choice. x^3+y=0, x^2−y=0

Verified step by step guidance

1. Identify the equations given in the system: Equation 1: x^3 + y = 0, Equation 2: x^2 - y = 0.
2. Express y from Equation 2 in terms of x: y = x^2.
3. Substitute the expression for y from Equation 2 into Equation 1: x^3 + x^2 = 0.
4. Factor out the common term from the equation: x^2(x + 1) = 0.
5. Solve for x by setting each factor equal to zero: x^2 = 0 and x + 1 = 0. Solve these equations to find the values of x, and then substitute back to find the corresponding values of y.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations that share the same 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 both cubic and quadratic equations, which may require specific methods for solving.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This technique is particularly useful when one equation can be easily manipulated to express a variable in terms of the other, simplifying the process of finding solutions to the system.

Graphical Interpretation

Graphical interpretation involves plotting the equations on a coordinate plane to visually identify the points of intersection, which represent the solutions to the system. For nonlinear equations like those in this problem, the graphs may yield multiple intersection points, indicating multiple solutions or no solution at all, depending on their shapes.