In Exercises 1–18, solve each system by the substitution method. {xy=3x2+y2=10\begin{cases} xy = 3 \\ $$x^2$$ + $$y^2 = 10$$ \end{cases}{xy=3x2+y2=10

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 13

6장, 문제 13

In Exercises 1–18, solve each system by the substitution method. $\begin{cases}\) xy = 3 \\ x^2 + y^2 = 10 \(\end{cases}$

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Start with the given system of equations: $xy = 3$ and $x^{2} + y^{2} = 10$.

From the first equation $xy = 3$, solve for one variable in terms of the other. For example, solve for $y$: $y = \frac{3}{x}$.

Substitute the expression for $y$ into the second equation $x^{2} + y^{2} = 10$. This gives: $x^{2} + \left(\frac{3}{x}\right)^{2} = 10$.

Simplify the substituted equation: $x^{2} + \frac{9}{x^{2}} = 10$. To clear the denominator, multiply both sides of the equation by $x^{2}$, resulting in $x^{4} + 9 = 10x^{2}$.

Rewrite the equation as a quadratic in terms of $x^{2}$: $x^{4} - 10x^{2} + 9 = 0$. Let $u = x^{2}$, then solve the quadratic equation $u^{2} - 10u + 9 = 0$ for $u$.

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주요 개념

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

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 useful when one equation is already solved for a variable or can be easily manipulated.

Solving Nonlinear Systems

Nonlinear systems include equations where variables are raised to powers or multiplied together, such as xy=3 and x² + y²=10. Solving these requires careful algebraic manipulation, often leading to quadratic equations. Understanding how to handle nonlinear terms is essential to find all possible solutions.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically in the form ax² + bx + c = 0. When using substitution, the system often reduces to a quadratic equation, which can be solved using factoring, completing the square, or the quadratic formula. Recognizing and solving quadratics is key to finding the values of variables.

In Exercises 1–18, solve each system by the substitution method. {xy=3x2+y2=10\(\begin{cases}\) xy = 3 \\ x^2 + y^2 = 10 \(\end{cases}\){xy=3x2+y2=10