In Exercises 19–28, solve each system by the addition method. {x2+y2=13x2−y2=5\begin{cases} $$x^2$$ + $$y^2 = 13$$ \\ $$x^2$$ - $$y^2 = 5$$ \end{cases}{x2+y2=13x2−y2=5

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 19

6장, 문제 19

In Exercises 19–28, solve each system by the addition method. $\begin{cases}\) x^2 + y^2 = 13 \\ x^2 - y^2 = 5 \(\end{cases}$

검증된 단계별 안내

Identify the given system of equations: \\ $x^2 + y^2 = 13$ and $x^2 - y^2 = 5$.

Add the two equations together to eliminate $y^2\(: \\ $(x^2 + y^2) + (x^2 - y^2) = 13 + 5\).

Simplify the resulting equation: \\ $2x^2 = 18$.

Solve for $x^2\( by dividing both sides by 2: \\ $x^2 = 9\).

Substitute $x^2 = 9$ back into one of the original equations (for example, $x^2 + y^2 = 13$) to solve for $y^2$, then find $y$.

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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 the coefficients of a variable are opposites or can be made opposites by multiplication.

Solving Quadratic Equations

Quadratic equations involve variables raised to the second power. Solving them may require factoring, using the quadratic formula, or simplifying expressions after elimination. Recognizing and solving quadratic forms is crucial when variables appear squared in systems.

In Exercises 19–28, solve each system by the addition method. {x2+y2=13x2−y2=5\(\begin{cases}\) x^2 + y^2 = 13 \\ x^2 - y^2 = 5 \(\end{cases}\){x2+y2=13x2−y2=5