Domanda del libro di testo
In Exercises 39–45, graph each inequality. y ≤ x^2 - 1
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Ch. 5 - Systems of Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 6, Problema 43
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 sum of two numbers is 10 and their product is 24. Find the numbers.
Guida verificata passo dopo passo1
Define the variables: let \(x\) represent one number and \(y\) represent the other number.
Write the first equation based on the sum condition: \(x + y = 10\).
Write the second equation based on the product condition: \(x \times y = 24\).
Use the first equation to express one variable in terms of the other, for example, \(y = 10 - x\).
Substitute \(y = 10 - x\) into the second equation to get a quadratic equation: \(x(10 - x) = 24\), which simplifies to \(10x - x^2 = 24\). Rearrange it to standard form: \(x^2 - 10x + 24 = 0\). Then solve this quadratic equation using factoring, completing the square, or the quadratic formula to find the values of \(x\), and subsequently \(y\).
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System of Nonlinear Equations
A system of nonlinear equations consists of two or more equations involving variables where at least one equation is nonlinear. In this problem, the sum and product conditions form equations that must be solved simultaneously to find the values of x and y.
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Nonlinear Inequalities
Sum and Product of Two Numbers
The sum of two numbers x and y is expressed as x + y, and their product as x * y. These relationships help form equations based on given conditions, which are essential for setting up the system to solve for the unknown numbers.
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Solving Quadratic Equations
After expressing one variable in terms of the other using the sum equation, substituting into the product equation leads to a quadratic equation. Solving this quadratic using factoring, completing the square, or the quadratic formula yields the possible values of the numbers.
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06:08Solving Quadratic Equations by Factoring
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