Book cover for Elementary and Intermediate Algebra

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson

ISBN #9781111567682

5th Edition

9,862 Questions

Group icon
237,244 Students Helped

Homework Questions

Right arrow
Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section emphasizes that anyone can improve their math abilities through effective study strategies, such as identifying your learning style and taking thorough notes. The core mathematical content centers on solving equations using properties of equality, which include adding, subtracting, multiplying, or dividing both sides of an equation. By transforming equations into simpler equivalent forms and verifying solutions through substitution, students build a solid foundation in algebra.

Learning Objectives

1

Identify your learning style and use effective study strategies to enhance algebra success.

2

Understand and apply the four fundamental properties of equality (addition, subtraction, multiplication, division) to solve linear equations.

3

Differentiate between equations and expressions and recognize what constitutes a solution.

4

Transform complex equations into simpler, equivalent forms and verify solutions by substitution.

Key Concepts

CONCEPT

DEFINITION

Equation

A statement that two expressions are equal, usually containing an equals sign (=).

Expression

A combination of numbers, variables, and operations that does not include an equality sign.

Solution

A number that, when substituted for the variable in an equation, makes the equation true.

Properties of Equality

Rules that state performing the same operation on both sides of an equation does not change its solution. These include the addition, subtraction, multiplication, and division properties of equality.

Equivalent Equations

Equations that have the same solution set even though they may look different.

Example Problems

Example 1

Fill in the blanks. A statement indicating that two expressions are equal, such as $x+1=7,$ is called an____________

Example 2

Fill in the blanks. Any number that makes an equation true when substituted for the variable is said to ___________ the equation. Such numbers are called__________

Example 3

Fill in the blanks. To______________an equation means to find all values of the variable that make the equation true.

Example 4

Fill in the blanks. To solve an equation, we_______________ the variable on one side of the equation.

Example 5

Fill in the blanks. Equations with the same solutions are called___________________equations.
Scroll left
Scroll right

Step-by-Step Explanations

QUESTION

Solve the equation: x - 2 = 3.

STEP-BY-STEP ANSWER:

Step 1: Recognize that x is decreased by 2 on the left side.
Step 2: To undo the subtraction, add 2 to both sides of the equation: (x - 2) + 2 = 3 + 2.
Step 3: This simplifies to x = 5.
Final Answer: {5}

Using the Addition Property of Equality

QUESTION

Check whether 9 is a solution of the equation 3y - 1 = 2y - 7.

STEP-BY-STEP ANSWER:

Step 1: Substitute 9 for y in both expressions: Left side: 3(9) - 1 = 27 - 1 = 26. Right side: 2(9) - 7 = 18 - 7 = 11.
Step 2: Compare both sides: Since 26 ≠ 11, the statement is false.
Final Answer: 9 is not a solution.

Checking a Proposed Solution

QUESTION

Solve the equation: 2x = 80.

STEP-BY-STEP ANSWER:

Step 1: Notice that 2x means x is being multiplied by 2.
Step 2: To isolate x, divide both sides of the equation by 2: (2x)/2 = 80/2.
Step 3: Simplify to get x = 40.
Final Answer: {40}

Using the Division Property of Equality

Scroll left
Scroll right

Common Mistakes

  • Failing to perform the same operation on both sides of the equation, which breaks the equivalence.
  • Confusing an equation with an expression, leading to misunderstandings about when a value is a solution.
  • Not checking a solution by substituting it back into the original equation.
  • Not simplifying fractions or results to their simplest form, which can lead to interpretation errors.