Ch. 5 - Systems of Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

Problem 45

Chapter 6, Problem 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.

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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.

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Related Practice

Textbook Question

In Exercises 39–45, graph each inequality. y ≤ x^2 - 1

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Textbook Question

In Exercises 39–45, graph each inequality. y ≤ 2^x

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Textbook Question

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 equations. Solve the system and find the numbers. The sum of two numbers is 7. If one number is subtracted from the other, their difference is -1. Find the numbers.

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Textbook Question

Graph the solution set of each system of inequalities or indicate that the system has no solution.

$\begin{cases}\)y $\geq$ x^2 - 1 $\x$ - y $\geq$ -1\(\end{cases}$

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Textbook Question

Perform each long division and write the partial fraction decomposition of the remainder term. (x⁴-x²+2)/(x³-x²)

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Textbook Question

Graph the solution set of each system of inequalities or indicate that the system has no solution.

$\begin{cases}\)x + y > 4 $\x$ + y > -1\(\end{cases}$