In Exercises 19–28, solve each system by the addition method. {3x2+4y2−16=02x2−3y2−5=0\begin{cases} $$3x^2$$ + $$4y^2 - 16 = 0$$ \\ $$2x^2$$ - $$3y^2 - 5 = 0$$ \end{cases}{3x2+4y2−16=02x2−3y2−5=0

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 23

6장, 문제 23

In Exercises 19–28, solve each system by the addition method. $\begin{cases}\) 3x^2 + 4y^2 - 16 = 0 \\ 2x^2 - 3y^2 - 5 = 0 \(\end{cases}$

검증된 단계별 안내

First, write down the system of equations clearly: $3x^{2} + 4y^{2} - 16 = 0$ $2x^{2} - 3y^{2} - 5 = 0$

Rearrange each equation to isolate the constant on the right side: $3x^{2} + 4y^{2} = 16$ $2x^{2} - 3y^{2} = 5$

To use the addition method, aim to eliminate one variable by making the coefficients of either $x^{2}$ or $y^{2}$ opposites. Multiply the first equation by 3 and the second equation by 4 to align the coefficients of $y^{2}$: $3(3x^{2} + 4y^{2}) = 3(16)$ which gives $9x^{2} + 12y^{2} = 48$ $4(2x^{2} - 3y^{2}) = 4(5)$ which gives $8x^{2} - 12y^{2} = 20$

Add the two new equations to eliminate $y^{2}$: $(9x^{2} + 12y^{2}) + (8x^{2} - 12y^{2}) = 48 + 20$ This simplifies to $17x^{2} = 68$

Solve for $x^{2}$ by dividing both sides by 17: $x^{2} = \frac{68}{17}$ Once you find $x^{2}$, substitute this value back into one of the original rearranged equations to solve for $y^{2}$.

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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 manipulate these systems is essential for finding their solutions.

Addition Method (Elimination Method)

The addition method involves adding or subtracting equations to eliminate one variable, simplifying the system to a single equation with one variable. This technique is useful when equations are aligned to cancel terms, making it easier to solve for the remaining variable.

Quadratic Equations in Two Variables

Quadratic equations involve variables raised to the second power, such as x² and y². When solving systems with quadratic terms, it is important to recognize the nature of the equations and apply appropriate algebraic methods, as solutions may include multiple or no real values.

In Exercises 19–28, solve each system by the addition method. {3x2+4y2−16=02x2−3y2−5=0\(\begin{cases}\) 3x^2 + 4y^2 - 16 = 0 \\ 2x^2 - 3y^2 - 5 = 0 \(\end{cases}\){3x2+4y2−16=02x2−3y2−5=0