In Exercises 29–42, solve each system by the method of your choice. {x2+4y2=20x+2y=6\begin{cases} $$x^2$$ + $$4y^2 = 20$$ \\ x + 2y = 6 \end{cases}{x2+4y2=20x+2y=6

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 33

6장, 문제 33

In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}\) x^2 + 4y^2 = 20 \\ x + 2y = 6 \(\end{cases}$

검증된 단계별 안내

Identify the system of equations: $x^2 + 4y^2 = 20$ and $x + 2y = 6$.

From the linear equation $x + 2y = 6$, solve for one variable in terms of the other. For example, solve for $x$: $x = 6 - 2y$.

Substitute the expression for $x$ into the first equation $x^2 + 4y^2 = 20$. This gives: $(6 - 2y)^2 + 4y^2 = 20$.

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

Solve the quadratic equation for $y$, then substitute each $y$ value back into $x = 6 - 2y$ to find the corresponding $x$ values.

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주요 개념

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

Systems of Equations

A system of equations consists of two or more equations with the same variables. The solution is the set of values that satisfy all equations simultaneously. Understanding how to interpret and solve these systems is fundamental in algebra.

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.

Solving Nonlinear Equations

Nonlinear equations, such as those involving squares or other powers, require special techniques like factoring, using the quadratic formula, or isolating terms. Recognizing and solving nonlinear equations is essential when working with systems that include curves or conic sections.

In Exercises 29–42, solve each system by the method of your choice. {x2+4y2=20x+2y=6\(\begin{cases}\) x^2 + 4y^2 = 20 \\ x + 2y = 6 \(\end{cases}\){x2+4y2=20x+2y=6