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

문제 35

6장, 문제 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}$

검증된 단계별 안내

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 $y = x^2$.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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