In Exercises 29–42, solve each system by the method of your choice. {3x2+4y2=162x2−3y2=5\begin{cases} $$3x^2$$ + $$4y^2 = 16$$ \\ $$2x^2$$ - $$3y^2 = 5$$ \end{cases}{3x2+4y2=162x2−3y2=5

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 29

6장, 문제 29

In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}\) 3x^2 + 4y^2 = 16 \\ 2x^2 - 3y^2 = 5 \(\end{cases}$

검증된 단계별 안내

Identify the system of equations to solve: \[3x^2 + 4y^2 = 16\] \[2x^2 - 3y^2 = 5\]

To solve the system, notice both equations involve $x^2$ and $y^2$. Let’s introduce new variables to simplify: Let \[a = x^2\] and \[b = y^2\]. Then rewrite the system as: \[3a + 4b = 16\] \[2a - 3b = 5\]

Solve the system of linear equations in terms of $a$ and $b$. For example, use the method of substitution or elimination: - Multiply one or both equations to align coefficients, - Then add or subtract the equations to eliminate one variable, - Solve for the remaining variable.

Once you find the values of $a$ and $b$, recall that $a = x^2$ and $b = y^2$. To find $x$ and $y$, take the square root of each value: \[x = \pm \sqrt{a}\] \[y = \pm \sqrt{b}\]

Check each pair $(x, y)$ in the original equations to verify the solutions, since squaring can introduce extraneous solutions.

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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 the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to interpret and set up systems is essential before applying any solving method.

Solving Systems by Substitution or Elimination

These are common methods to solve systems. Substitution involves solving one equation for a variable and substituting into the other. Elimination involves adding or subtracting equations to eliminate a variable. Both methods can be adapted for nonlinear systems like the given quadratic equations.

Handling Quadratic Equations in Systems

When systems include quadratic terms, solutions may be multiple or complex. Recognizing how to manipulate and combine quadratic expressions, such as factoring or isolating terms, is crucial. This helps in reducing the system to simpler forms or single-variable equations.

In Exercises 29–42, solve each system by the method of your choice. {3x2+4y2=162x2−3y2=5\(\begin{cases}\) 3x^2 + 4y^2 = 16 \\ 2x^2 - 3y^2 = 5 \(\end{cases}\){3x2+4y2=162x2−3y2=5