Book cover for Beginning Algebra

Beginning Algebra

Margaret L. Lial, John Hornsby

ISBN #9780321673480

11th Edition

5,012 Questions

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99,418 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section focuses on systems of linear equations and demonstrates how to determine whether an ordered pair is a solution, solve systems by graphing, and recognize special systems that lack a unique solution. The graphical method offers visual insight, and verifying solutions by substitution is critical to ensure accuracy. Understanding these concepts builds a foundation for more advanced topics in algebra and practical problem-solving scenarios.

Learning Objectives

1

Describe what a system of linear equations is and explain the concept of a solution set.

2

Determine whether a given ordered pair satisfies a system of linear equations.

3

Solve systems of linear equations by graphing the lines and identifying intersection points.

4

Identify and differentiate between special systems that have no solution or infinitely many solutions.

Key Concepts

CONCEPT

DEFINITION

System of Linear Equations

A collection of two or more linear equations involving the same set of variables.

Solution (Satisfies the Equation)

An ordered pair (or tuple) that makes every equation in the system true when substituted for the variables.

Graphical Method

A method for solving systems of equations by plotting each equation on the coordinate plane and identifying the point(s) where the graphs intersect.

Special Systems

Systems where the lines are either parallel (resulting in no solution) or coincident (resulting in infinitely many solutions).

Set-Builder Notation

A concise way of specifying a set by stating the properties that its members must satisfy, often used to denote an infinite solution set.

Example Problems

Example 1

Which ordered pair could not be a solution of the system graphed? Why is it the only valid choice? A. $(-4,-4)$ B. $(-2,2)$ C. $(-4,4)$ D. $(-3,3)$ (GRAPH CAN'T COPY)

Example 2

Concept Check Which ordered pair could be a solution of the system graphed? Why is it the only valid choice? A. $(2,0)$ B. $(0,2)$ C. $(-2,0)$ D. $(0,-2)$ (GRAPH CAN'T COPY)

Example 3

Decide whether the given ordered pair is a solution of the given system. $$ \begin{aligned} &\begin{array}{c} {(2,-3)} \\ {x+y=-1} \end{array}\\ &2 x+5 y=19 \end{aligned} $$

Example 4

Decide whether the given ordered pair is a solution of the given system. $$ \begin{array}{c} {(4,3)} \\ {x+2 y=10} \\ {3 x+5 y=3} \end{array} $$

Example 5

Decide whether the given ordered pair is a solution of the given system. $$ \begin{aligned} &(-1,-3)\\ &3 x+5 y=-18\\ &4 x+2 y=-10 \end{aligned} $$
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Step-by-Step Explanations

QUESTION

Given the system: 2x + y = 12 and 3x - y = 13, determine if (4, 4) is a solution.

STEP-BY-STEP ANSWER:

Step 1: Substitute x = 4 and y = 4 into the first equation: 2(4) + 4 = 8 + 4 = 12. Since the left side equals 12, the first equation is satisfied.
Step 2: Substitute x = 4 and y = 4 into the second equation: 3(4) - 4 = 12 - 4 = 8, which does not equal 13. Therefore, the second equation is not satisfied.
Final Answer: (4, 4) is not a solution of the system.

Checking an Ordered Pair as a Solution

QUESTION

Solve the system by graphing: 3x - y = -5 and 2x + 3y = 4.

STEP-BY-STEP ANSWER:

Step 1: Rewrite each equation in slope-intercept form if necessary. For example, 3x - y = -5 can be rewritten as y = 3x + 5.
Step 2: Identify and plot at least two points for each line using intercepts or other convenient values.
Step 3: Draw the lines on the coordinate plane and identify the intersection point, which is the solution.
Step 4: Verify the intersection point by substituting its coordinates into both original equations.
Final Answer: The intersection point that satisfies both equations is the solution to the system.

Solving by Graphing

QUESTION

Determine the solution set for the system: 2x + y = 8 and 2x + y = 2.

STEP-BY-STEP ANSWER:

Step 1: Note that both equations have the same left-hand side but different right-hand sides.
Step 2: Graph the first equation, 2x + y = 8, and then the second equation, 2x + y = 2.
Step 3: Observe that the lines are parallel (they have the same slope) and never intersect.
Step 4: Since no intersection exists, no ordered pair satisfies both equations simultaneously.
Final Answer: The system has no solution; the solution set is empty, often represented as ∅ or {}.

Solving Special Systems by Graphing

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Common Mistakes

  • Failing to substitute the ordered pair into every equation in the system.
  • Mixing up variables during substitution, leading to incorrect evaluations.
  • Relying solely on approximate graph intersections without proper verification by substitution.
  • Incorrectly identifying parallel lines as having a potential intersection due to drawing errors.
  • Not recognizing when two equations represent the same line (infinitely many solutions).