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

Problem 19

Chapter 6, Problem 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}$

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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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

Verified video answer for a similar problem:

Key Concepts

Systems of Equations

Addition Method (Elimination Method)

Solving Quadratic Equations

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}$