Textbook Question
Solve each system in Exercises 5–18.
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Ch. 5 - Systems of Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- 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
Chapter 6, Problem 9
In Exercises 1–18, solve each system by the substitution method.
Verified step by step guidance1
Start with the given system of equations: \(xy = 6\) and \(2x - y = 1\).
From the second equation \(2x - y = 1\), solve for \(y\) in terms of \(x\): \(y = 2x - 1\).
Substitute the expression for \(y\) into the first equation \(xy = 6\): \(x(2x - 1) = 6\).
Expand and rewrite the equation: \(2x^2 - x = 6\).
Rearrange the equation to standard quadratic form: \(2x^2 - x - 6 = 0\), then solve for \(x\) using the quadratic formula or factoring.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
System 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.
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Introduction to Systems of Linear Equations
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 easily solved for one variable.
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Solving Nonlinear Equations
When a system includes nonlinear equations, such as xy = 6, solving requires careful algebraic manipulation. After substitution, the resulting equation may be quadratic or another nonlinear form, requiring techniques like factoring or the quadratic formula to find solutions.
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Related Practice
Textbook Question
An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.
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Textbook Question
A chemist needs to mix a solution that is 34% silver nitrate with one that is 4% silver nitrate to obtain 100 milliliters of a mixture that is 7% silver nitrate. How many milliliters of each of the solutions must be used?
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Textbook Question
In Exercises 5–18, solve each system by the substitution method.
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Textbook Question
The perimeter of a table tennis top is 28 feet. The difference between 4 times the length and 3 times the width is 21 feet. Find the dimensions.
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Textbook Question
In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. (7x2 -9x+3)/(x2+7)2
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