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 31

Chapter 6, Problem 31

In Exercises 31–42, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. x = 9-2y x + 2y = 13

Verified step by step guidance

Start with the given system of equations:

x = 9 - 2 y

and

x + 2 y = 13

Since the first equation already expresses

x

in terms of

y

, substitute

x = 9 - 2y

into the second equation.

After substitution, the second equation becomes

(9 - 2y) + 2y = 13

. Simplify this equation by combining like terms.

Analyze the simplified equation to determine if it is true for all values of

y

, no values, or specific values. This will help identify if the system has one solution, no solution, or infinitely many solutions.

Based on the result, express the solution set using set notation. If there is one solution, write it as an ordered pair

(x, y)

. If no solution exists, state the empty set

\(\varnothing\)

. If infinitely many solutions exist, express the solution set parametrically.

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Key Concepts

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

Solving Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. Methods include substitution, elimination, and graphing, each useful depending on the system's form.

Types of Solutions for Systems

Systems of linear equations can have one solution (consistent and independent), no solution (inconsistent), or infinitely many solutions (dependent). Identifying the type involves analyzing the equations for contradictions or equivalences, which affects how the solution set is expressed.

Set Notation for Solution Sets

Set notation is a concise way to represent all solutions of a system. For a unique solution, it lists the ordered pair; for infinitely many solutions, it describes the solution set using parameters; for no solution, the set is empty. This notation clearly communicates the solution structure.

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Related Practice

Textbook Question

In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}\)x^2 + 4y^2 = 20 $\x$ + 2y = 6\(\end{cases}$

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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 + 2y $\leq$ 4 $\y$ $\geq$ x - 3\(\end{cases}$

Textbook Question

Write the partial fraction decomposition of each rational expression. 5x²+6x+3/(x + 1)(x² + 2x + 2)

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Textbook Question

In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}\)2x^2 + y^2 = 18 $\xy$ = 4\(\end{cases}$

Textbook Question

In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}\)3x^2 + 4y^2 = 16 \\2x^2 - 3y^2 = 5\(\end{cases}$

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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 > 2x - 3 $\y$ < -x + 6\(\end{cases}$