In Exercises 1–18, solve each system by the substitution method. {x+y=1x2+xy−y2=−5\begin{cases} x + y = 1 \\ $$x^2 + xy$$ - $$y^2 = -5$$ \end{cases}{x+y=1x2+xy−y2=−5

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 15

6장, 문제 15

In Exercises 1–18, solve each system by the substitution method. $\begin{cases}\) x + y = 1 \\ x^2 + xy - y^2 = -5 \(\end{cases}$

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Start with the given system of equations: $x + y = 1$ and $x^2 + xy - y^2 = -5$.

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

Substitute the expression for $x$ from step 2 into the second equation: $(1 - y)^2 + (1 - y) y - y^2 = -5$.

Expand and simplify the substituted equation by applying algebraic operations such as expanding squares and distributing terms: Expand $(1 - y)^2$ to $1 - 2y + y^2$, then combine like terms carefully.

After simplification, you will get a quadratic equation in terms of $y$. Solve this quadratic equation using factoring, completing the square, or the quadratic formula to find the possible values of $y$. Then, substitute these values back into $x = 1 - y$ to find the corresponding $x$ values.

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주요 개념

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

Substitution Method

The substitution method is a technique used to solve systems of equations. It involves solving one equation for one variable and then substituting that expression into the other equation. This method simplifies the system, allowing for easier solving of the equations, especially when one equation is linear.

Linear Equations

Linear equations are algebraic expressions that represent straight lines when graphed. They can be written in the form Ax + By = C, where A, B, and C are constants. In the given system, the equation x + y = 1 is linear, which allows for straightforward manipulation and substitution.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically in the form ax^2 + bx + c = 0. The equation x^2 + xy - y^2 = -5 contains both linear and quadratic terms, making it essential to recognize how to handle these types of equations when substituting values from one equation into another.

In Exercises 1–18, solve each system by the substitution method. {x+y=1x2+xy−y2=−5\(\begin{cases}\) x + y = 1 \\ x^2 + xy - y^2 = -5 \(\end{cases}\){x+y=1x2+xy−y2=−5