In Exercises 19–28, solve each system by the addition method. {x2−4y2=−73x2+y2=31\begin{cases} $$x^2$$ - $$4y^2 = -7$$ \\ $$3x^2$$ + $$y^2 = 31$$ \end{cases}{x2−4y2=−73x2+y2=31

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 21

6장, 문제 21

In Exercises 19–28, solve each system by the addition method. $\begin{cases}\) x^2 - 4y^2 = -7 \\ 3x^2 + y^2 = 31 \(\end{cases}$

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Identify the system of equations: $x^2 - 4y^2 = -7$ $3x^2 + y^2 = 31$

To use the addition method, first align the equations so that adding or subtracting them will eliminate one variable. Notice the variables are $x^2$ and $y^2$, so treat $x^2$ and $y^2$ as separate terms.

Multiply the second equation by 4 to match the coefficient of $y^2$ in the first equation: $4(3x^2 + y^2) = 4(31)$ which gives $12x^2 + 4y^2 = 124$

Now add the first equation and the new equation: $(x^2 - 4y^2) + (12x^2 + 4y^2) = -7 + 124$ Simplify the left side to eliminate $y^2$ terms and combine like terms.

Solve the resulting equation for $x^2$, then substitute this value back into one of the original equations to find $y^2$. Finally, take square roots to find $x$ and $y$ values, remembering to consider both positive and negative roots.

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주요 개념

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Systems of Equations

A system of equations consists of two or more equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to interpret and manipulate these systems is essential for finding their solutions.

Addition Method (Elimination Method)

The addition method involves adding or subtracting equations to eliminate one variable, simplifying the system to a single equation with one variable. This technique is useful when equations are arranged so that adding them cancels out a variable, making it easier to solve.

Handling Quadratic Terms in Systems

When systems include quadratic terms like x² and y², solving requires careful manipulation, such as combining like terms or substitution after elimination. Recognizing the structure of quadratic expressions helps in applying the addition method effectively to find all possible solutions.

In Exercises 19–28, solve each system by the addition method. {x2−4y2=−73x2+y2=31\(\begin{cases}\) x^2 - 4y^2 = -7 \\ 3x^2 + y^2 = 31 \(\end{cases}\){x2−4y2=−73x2+y2=31