In Exercises 29–42, solve each system by the method of your choice. {x2+y2+3y=222x+y=−1\begin{cases} $$x^2$$ + $$y^2 + 3y = 22$$ \\ 2x + y = -1 \end{cases}{x2+y2+3y=222x+y=−1

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 41

6장, 문제 41

In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}\) x^2 + y^2 + 3y = 22 \\ 2x + y = -1 \(\end{cases}$

검증된 단계별 안내

Start by examining the given system of equations: $x^2 + y^2 + 3y = 22$ and $2x + y = -1$.

From the linear equation $2x + y = -1$, solve for $y$ in terms of $x$: $y = -1 - 2x$.

Substitute the expression for $y$ into the first equation to eliminate $y$: $x^2 + (-1 - 2x)^2 + 3(-1 - 2x) = 22$.

Expand and simplify the resulting equation to form a quadratic equation in terms of $x$ only.

Solve the quadratic equation for $x$, then substitute each solution back into $y = -1 - 2x$ to find the corresponding $y$ values.

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주요 개념

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

Solving Systems of Equations

A system of equations consists of two or more equations with the same variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. Methods include substitution, elimination, and graphing, each useful depending on the system's form.

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 effective when one equation is linear.

Quadratic Equations and Circles

The equation x² + y² + 3y = 22 represents a circle after completing the square for y. Understanding how to manipulate and solve quadratic equations is essential to find the points of intersection with the linear equation. This helps determine the system's solutions.

In Exercises 29–42, solve each system by the method of your choice. {x2+y2+3y=222x+y=−1\(\begin{cases}\) x^2 + y^2 + 3y = 22 \\ 2x + y = -1 \(\end{cases}\){x2+y2+3y=222x+y=−1