In Exercises 47–52, solve each system by the method of your choice. {3x2+1y2=75x2−2y2=−3\begin{cases} \frac{3}{$$x^2$$} + \frac{1}{$$y^2$$} = 7 \\ \frac{5}{$$x^2$$} - \frac{2}{$$y^2$$} = -3 \end{cases}{x23+y21=7x25−y22=−3

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 51

6장, 문제 51

In Exercises 47–52, solve each system by the method of your choice. $\begin{cases}\) $\frac{3}{x^2}$ + $\frac{1}{y^2}$ = 7 \\ $\frac{5}{x^2}$ - $\frac{2}{y^2}$ = -3 \(\end{cases}$

검증된 단계별 안내

Identify the system of equations: \[ \frac{3}{x^2} + \frac{1}{y^2} = 7 \] \[ \frac{5}{x^2} - \frac{2}{y^2} = -3 \]

To simplify the system, introduce new variables: let \[ a = \frac{1}{x^2} \quad \text{and} \quad b = \frac{1}{y^2} \] This transforms the system into linear equations in terms of $a$ and $b$.

Rewrite the system using the new variables: \[ 3a + b = 7 \] \[ 5a - 2b = -3 \]

Solve the linear system for $a$ and $b$ using either substitution or elimination method. For example, multiply the first equation to align coefficients and eliminate one variable.

Once you find values for $a$ and $b$, substitute back to find $x$ and $y$ by solving: \[ a = \frac{1}{x^2} \implies x^2 = \frac{1}{a} \] \[ b = \frac{1}{y^2} \implies y^2 = \frac{1}{b} \] Then take square roots to find $x$ and $y$, remembering to consider both positive and negative roots.

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.

Systems of Equations

A system of equations consists of two or more equations with multiple variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to manipulate and solve these systems is fundamental in algebra.

Substitution and Elimination Methods

These are common techniques for solving systems of equations. Substitution involves solving one equation for a variable and substituting into the other, while elimination involves adding or subtracting equations to eliminate a variable. Choosing the appropriate method simplifies solving complex systems.

Handling Rational Expressions and Variables in Denominators

When variables appear in denominators, it is important to rewrite the equations to avoid division by zero and simplify the system. This often involves substituting new variables for expressions like 1/x² or 1/y², turning the system into a more manageable form.

In Exercises 47–52, solve each system by the method of your choice. {3x2+1y2=75x2−2y2=−3\(\begin{cases}\) \(\frac{3}{x^2}\) + \(\frac{1}{y^2}\) = 7 \\ \(\frac{5}{x^2}\) - \(\frac{2}{y^2}\) = -3 \(\end{cases}\){x23+y21=7x25−y22=−3

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Systems of Equations

Substitution and Elimination Methods

Handling Rational Expressions and Variables in Denominators

In Exercises 47–52, solve each system by the method of your choice. $\begin{cases}\) \(\frac{3}{x^2}\) + \(\frac{1}{y^2}\) = 7 \\ \(\frac{5}{x^2}\) - \(\frac{2}{y^2}\) = -3 \(\end{cases}$