In Exercises 1–18, solve each system by the substitution method. {x+y=1(x−1)2+(y+2)2=10\begin{cases} x + y = 1 \\ (x - $$1)^2 + (y$$ + $$2)^2 = 10$$ \end{cases}{x+y=1(x−1)2+(y+2)2=10

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 17

6장, 문제 17

In Exercises 1–18, solve each system by the substitution method. $\begin{cases}\) x + y = 1 \\ (x - 1)^2 + (y + 2)^2 = 10 \(\end{cases}$

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Start with the given system of equations: $x + y = 1$ and $(x - 1)^2 + (y + 2)^2 = 10$.

From the first equation, solve for one variable in terms of the other. For example, express $y$ as $y = 1 - x$.

Substitute the expression for $y$ into the second equation to replace $y$ with $1 - x$. This gives: $(x - 1)^2 + ((1 - x) + 2)^2 = 10$.

Simplify the equation by expanding the squares and combining like terms to form a quadratic equation in terms of $x$.

Solve the quadratic equation for $x$, then substitute each solution back into $y = 1 - x$ to find the corresponding $y$ values.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

System 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.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, making it easier to solve. It is especially useful when one equation is already solved or easily solved for a variable.

Solving Quadratic Equations

When substitution leads to a quadratic equation, techniques such as factoring, completing the square, or using the quadratic formula are used to find solutions. Recognizing and solving quadratic equations is crucial for systems involving nonlinear equations like circles or parabolas.

In Exercises 1–18, solve each system by the substitution method. {x+y=1(x−1)2+(y+2)2=10\(\begin{cases}\) x + y = 1 \\ (x - 1)^2 + (y + 2)^2 = 10 \(\end{cases}\){x+y=1(x−1)2+(y+2)2=10