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 the use of the addition and subtraction properties of equality to solve linear equations by subtracting or adding the same number on both sides. Key strategies include combining like terms, using the additive inverse to isolate the variable, and checking solutions through substitution. Special care must be taken when applying the distributive property to avoid sign errors. Understanding these fundamentals aids in solving more complex problems and analyzing real-world scenarios.

Learning Objectives

1

Apply the addition property of equality to transform and solve linear equations.

2

Combine like terms and use the additive inverse to isolate variables.

3

Utilize subtraction from both sides to simplify equations and generalize solutions.

4

Check solutions by substituting back into the original equation.

5

Correctly apply the distributive property when solving equations.

Key Concepts

CONCEPT

DEFINITION

Addition Property of Equality

If the same quantity is added to both sides of an equation, the equality is preserved.

Subtraction Property of Equality

If the same quantity is subtracted from both sides of an equation, the equation remains balanced.

Additive Inverse

For any number x, the additive inverse is -x because x + (-x) = 0.

Combining Like Terms

The process of simplifying algebraic expressions by summing coefficients of similar terms.

Distributive Property

A property that allows one to multiply a sum by multiplying each addend separately by a common factor, i.e., a(b + c) = ab + ac.

Example Problems

Example 1

Decide whether each of the following is an expression or an equation. If it is an expression, simplify it. If it is an equation, solve it. $\begin{array}{ll}{\text { (a) } 5 x+8-4 x+7} & {\text { (b) }-6 y+12+7 y-5} \\ {\text { (c) } 5 x+8-4 x=7} & {\text { (d) }-6 y+12+7 y=-5}\end{array}$

Example 2

Which pairs of equations are equivalent equations? $\begin{array}{ll}{\text { A. } x+2=6 \text { and } x=4} & {\text { B. } 10-x=5 \text { and } x=-5} \\ {\text { C. } x+3=9 \text { and } x=6} & {\text { D. } 4+x=8 \text { and } x=-4}\end{array}$

Example 3

Which of the following are not linear equations in one variable? $\begin{array}{llll}{\text { A. } x^{2}-5 x+6=0} & {\text { B. } x^{3}=x} & {\text { C. } 3 x-4=0} & {\text { D. } 7 x-6 x=3+9 x}\end{array}$

Example 4

Explain how to check a solution of an equation.

Example 5

Solve each equation, and check your solution. $$ x-3=9 $$
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Step-by-Step Explanations

QUESTION

What happens if we begin solving Example 4 by subtracting a number from each side of the equation?

STEP-BY-STEP ANSWER:

Step 1: Begin with the original equation from Example 4.
Step 2: Subtract the same number (e.g., 17) from both sides of the equation. This uses the subtraction property of equality to maintain balance.
Step 3: Combine like terms on each side. The subtraction gives an equation where the additive inverse of x is obtained.
Step 4: Recognize that if the additive inverse of x equals a value (like -17), then x must be the opposite sign, meaning x = 17.
Final Answer: After subtraction and combining like terms, the value of x is found to be 17.

Example 4: Subtracting from Both Sides

QUESTION

How do you solve the equation 6x - 8 = 12 + 5x by first using the addition property?

STEP-BY-STEP ANSWER:

Step 1: Start with the equation: 6x - 8 = 12 + 5x.
Step 2: Subtract 5x from both sides to bring variable terms together: 6x - 5x - 8 = 12.
Step 3: Simplify the left side: x - 8 = 12.
Step 4: Add 8 to both sides to isolate the variable: x = 12 + 8.
Final Answer: x = 20.

Solving an Equation Using the Addition Property

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Common Mistakes

  • Failing to subtract or add the same number to both sides, which disrupts the balance of the equation.
  • Incorrectly combining like terms due to sign errors or not properly applying the additive inverse.
  • Overlooking the need to check solutions by substituting back into the original equation.
  • Misapplying the distributive property, which may lead to errors with signs.