In Exercises 47–52, solve each system by the method of your choice. {2x2+xy=6x2+2xy=0\begin{cases} $$2x^2 + xy = 6$$ \\ $$x^2 + 2xy = 0$$ \end{cases}{2x2+xy=6x2+2xy=0

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 47

6장, 문제 47

In Exercises 47–52, solve each system by the method of your choice. $\begin{cases}\) 2x^2 + xy = 6 \\ x^2 + 2xy = 0 \(\end{cases}$

검증된 단계별 안내

Start by writing down the system of equations clearly: \$2x^{2} + xy = 6$ and $x^{2} + 2xy = 0$.

From the second equation $x^{2} + 2xy = 0$, factor out the common term $x$: $x(x + 2y) = 0$.

Set each factor equal to zero to find possible cases: either $x = 0$ or $x + 2y = 0$.

For the case $x = 0$, substitute into the first equation \$2(0)^{2} + 0 imes y = 6$ and check if it holds true.

For the case $x + 2y = 0$, solve for $y$ as $y = -\frac{x}{2}$, then substitute this expression for $y$ into the first equation $2x^{2} + x\left(-\frac{x}{2}\right) = 6$ to find values of $x$.

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주요 개념

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

Systems of Nonlinear Equations

A system of nonlinear equations involves two or more equations with variables raised to powers other than one or multiplied together. Solving such systems requires finding values for the variables that satisfy all equations simultaneously, often involving substitution or elimination methods adapted for nonlinear terms.

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, which can be solved using algebraic techniques, especially useful when one equation can be easily rearranged.

Factoring and Quadratic Equations

Factoring is a technique to rewrite expressions as products of simpler expressions, often used to solve quadratic equations. In systems involving quadratic terms, factoring helps find roots or solutions by setting each factor equal to zero, simplifying the process of solving nonlinear systems.

In Exercises 47–52, solve each system by the method of your choice. {2x2+xy=6x2+2xy=0\(\begin{cases}\) 2x^2 + xy = 6 \\ x^2 + 2xy = 0 \(\end{cases}\){2x2+xy=6x2+2xy=0