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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Blitzer 8th EditionCh. 5 - Systems of Equations and Inequalities문제 45
6장, 문제 45

In Exercises 43–46, let x represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The difference between the squares of two numbers is 3. Twice the square of the first number increased by the square of the second number is 9. Find the numbers.

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1
Define the variables: let \(x\) be the first number and \(y\) be the second number.
Translate the first condition "The difference between the squares of two numbers is 3" into an equation: \(x^{2} - y^{2} = 3\).
Translate the second condition "Twice the square of the first number increased by the square of the second number is 9" into an equation: \(2x^{2} + y^{2} = 9\).
You now have the system of nonlinear equations: \[ \begin{cases} x^{2} - y^{2} = 3 \\ 2x^{2} + y^{2} = 9 \end{cases} \] Use substitution or elimination to solve this system. For example, add the two equations to eliminate \(y^{2}\) or express \(y^{2}\) from one equation and substitute into the other.
After finding \(x^{2}\) and \(y^{2}\), take the square roots to find possible values for \(x\) and \(y\). Remember to consider both positive and negative roots since squaring eliminates sign information.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념

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

Formulating Systems of Nonlinear Equations

This involves translating word problems into mathematical equations where variables represent unknown quantities. In this case, expressions like 'difference between squares' and 'twice the square' are converted into algebraic equations involving x and y. Understanding how to set up these equations correctly is essential for solving the problem.
추천 영상:
3:21
Nonlinear Inequalities

Solving Systems of Nonlinear Equations

Unlike linear systems, nonlinear systems involve variables raised to powers or multiplied together. Methods such as substitution or elimination are used, often requiring manipulation of one equation to express one variable in terms of the other before solving. Recognizing the nonlinear nature guides the choice of solution strategy.
추천 영상:
5:48
Solving Systems of Equations - Substitution

Properties of Squares and Differences of Squares

Understanding how to work with squares and their differences is crucial. For example, the difference of squares formula (a² - b² = (a - b)(a + b)) can simplify expressions or help in factoring. Recognizing these properties aids in both forming and solving the equations efficiently.
추천 영상:
02:20
Imaginary Roots with the Square Root Property
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