In Exercises 29–42, solve each system by the method of your choice. {x3+y=0x2−y=0\begin{cases} $$x^3 + y = 0$$ \\ $$x^2 - y = 0$$ \end{cases}{x3+y=0x2−y=0

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

Problem 35

Chapter 6, Problem 35

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}$

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Start with the given system of equations: $x^3 + y = 0$ and $x^2 - y = 0$.

From the second equation, express $y$ in terms of $x$: $y = x^2$.

Substitute $y = x^2$ into the first equation to eliminate $y$: $x^3 + x^2 = 0$.

Factor the resulting equation: $x^2(x + 1) = 0$.

Set each factor equal to zero and solve for $x$: $x^2 = 0$ or $x + 1 = 0$, then find corresponding $y$ values using $y = x^2$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Related Practice

Textbook Question

In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}\)x^2 + 4y^2 = 20 $\x$ + 2y = 6\(\end{cases}$

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Textbook Question

In Exercises 31–42, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. 3x - 2y = − 5 4x + y = 8

Textbook Question

Write the partial fraction decomposition of each rational expression. x+4/x² (x²+4)

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Textbook Question

Graph the solution set of each system of inequalities or indicate that the system has no solution.

$\begin{cases}\)x $\leq$ 2 $\y$ $\geq$ -1\(\end{cases}$

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Textbook Question

Write the partial fraction decomposition of each rational expression. 6x²-x+1/(x³ + x²+ x +1)

In Exercises 29–42, solve each system by the method of your choice. {x3+y=0x2−y=0\(\begin{cases}\)x^3 + y = 0 \(\x\)^2 - y = 0\(\end{cases}\){x3+y=0x2−y=0

Verified video answer for a similar problem:

Key Concepts

Solving Systems of Equations

Substitution Method

Nonlinear Equations and Polynomial Functions

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}$