In Exercises 19–28, solve each system by the addition method. {x2+y2=25(x−8)2+y2=41\begin{cases} $$x^2$$ + $$y^2 = 25$$ \\ (x - $$8)^2$$ + $$y^2 = 41$$ \end{cases}{x2+y2=25(x−8)2+y2=41

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 25

6장, 문제 25

In Exercises 19–28, solve each system by the addition method. $\begin{cases}\) x^2 + y^2 = 25 \\ (x - 8)^2 + y^2 = 41 \(\end{cases}$

검증된 단계별 안내

Identify the system of equations: $x^2 + y^2 = 25$ and $(x - 8)^2 + y^2 = 41$.

Expand the second equation: $(x - 8)^2 + y^2 = x^2 - 16x + 64 + y^2 = 41$.

Rewrite the expanded second equation as $x^2 - 16x + 64 + y^2 = 41$.

Subtract the first equation $x^2 + y^2 = 25$ from the expanded second equation to eliminate $y^2$ and $x^2$: $(x^2 - 16x + 64 + y^2) - (x^2 + y^2) = 41 - 25$.

Simplify the resulting equation to isolate $x$ and solve for its value(s).

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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. In this problem, the system involves two equations representing circles, and solving the system means finding their points of intersection.

Addition Method (Elimination Method)

The addition method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the other. This method is typically used for linear systems, but can be adapted here by expanding and simplifying the given equations to linear form before elimination.

Equations of Circles and Their Expansion

Each equation represents a circle in standard form. Expanding the squared terms converts the equations into polynomial form, which can then be manipulated algebraically. Understanding how to expand and simplify these equations is essential to apply the addition method effectively.

In Exercises 19–28, solve each system by the addition method. {x2+y2=25(x−8)2+y2=41\(\begin{cases}\) x^2 + y^2 = 25 \\ (x - 8)^2 + y^2 = 41 \(\end{cases}\){x2+y2=25(x−8)2+y2=41