In Exercises 1–18, solve each system by the substitution method.{x2+y2=25x−y=1\begin{cases} $$x^2$$ + $$y^2 = 25$$ \\ x - y = 1 \end{cases}{x2+y2=25x−y=1

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 7

6장, 문제 7

In Exercises 1–18, solve each system by the substitution method. $\begin{cases}\) x^2 + y^2 = 25 \\ x - y = 1 \(\end{cases}$

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Start with the given system of equations: $x^2 + y^2 = 25$ and $x - y = 1$.

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

Substitute the expression for $x$ into the first equation to replace $x$ with $y + 1$, resulting in $(y + 1)^2 + y^2 = 25$.

Expand the squared term and combine like terms to form a quadratic equation in terms of $y$: $y^2 + 2y + 1 + y^2 = 25$.

Simplify the equation and solve the quadratic for $y$. Then, use the values of $y$ to find the corresponding $x$ values using $x = y + 1$.

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주요 개념

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

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 solvable for one variable.

Solving Quadratic Equations

When substitution leads to a quadratic equation, you must solve it using methods such as factoring, completing the square, or the quadratic formula. Quadratic equations can have two, one, or no real solutions, which correspond to the number of intersection points between the curves represented by the system.

Systems of Nonlinear Equations

A system involving equations like x² + y² = 25 (a circle) and x - y = 1 (a line) is nonlinear. Understanding the geometric interpretation helps: solutions correspond to intersection points of the circle and line. Such systems may have zero, one, or two solutions depending on how the line intersects the circle.

In Exercises 1–18, solve each system by the substitution method.{x2+y2=25x−y=1\(\begin{cases}\) x^2 + y^2 = 25 \\ x - y = 1 \(\end{cases}\){x2+y2=25x−y=1