Solve each system by the substitution method. {x+y=2y=x2−4\begin{cases} x + y = 2 \\ y = $$x^2 - 4$$ \end{cases}{x+y=2y=x2−4

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 1

6장, 문제 1

Solve each system by the substitution method.
$\begin{cases}\) x + y = 2 \\ y = x^2 - 4 \(\end{cases}$

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Start with the given system of equations: $x + y = 2$ and $y = x^{2} - 4$.

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

Substitute this expression for $y$ into the second equation: $2 - x = x^{2} - 4$.

Rewrite the equation to set it equal to zero: $x^{2} - 4 - (2 - x) = 0$, which simplifies to $x^{2} - 4 - 2 + x = 0$.

Combine like terms to get a quadratic equation: $x^{2} + x - 6 = 0$. This quadratic can now be solved using factoring, completing the square, or the quadratic formula.

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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 set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. In this problem, the system includes a linear and a quadratic equation involving x and y.

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. Here, y is expressed from the first equation and substituted into the second.

Solving Quadratic Equations

After substitution, the resulting equation is quadratic, which can be solved by factoring, completing the square, or using the quadratic formula. The solutions to the quadratic give possible values for x, which can then be used to find corresponding y values.

Solve each system by the substitution method. {x+y=2y=x2−4\(\begin{cases}\) x + y = 2 \\ y = x^2 - 4 \(\end{cases}\){x+y=2y=x2−4