Back To Course

High School Algebra II: Homework Help Resource26 chapters | 280 lessons | 1 flashcard set

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over 70,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Free 5-day trial
Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Joseph Vigil*

In this lesson, you'll learn what an algebraic expression is and what makes two algebraic expressions equivalent. You'll also see some examples of equivalent expressions.

An **algebraic expression** is a string of numbers, variables, mathematical operations, and possibly exponents. For example, 4*x* + 3 is a basic algebraic expression. Or, we could get a little more complex with 3*x*(2*x*^2 + 2*x* - 5) + 6*y*. Notice that both of these examples contain the previously listed elements of an algebraic expression: numbers, variables, and mathematical operations, and the second expression contains the optional exponent.

Also notice that there's no equal sign in an algebraic expression. If there were one, it would be an **equation**, or a number sentence that involves one or more variables. For example: 2*x* + 1 = 3. An equation has one specific solution or set of solutions that will make the number sentence true. In this case, the equation is a true number sentence when *x* = 1. There is one specific solution. In an expression, however, since there's no equal sign, variables are free to be variables. We're not looking for any specific values to stand in for them.

Let's take a look at equivalent expressions. What are they? As the name suggests, **equivalent expressions** are algebraic expressions that, although they look different, turn out to really be the same. And since they're the same, they will yield the same results no matter what numbers we substitute for their variables.

Let's consider this algebraic expression: 2(*x*^2 + *x*). If we substitute 1 for the variable, the expression equals 4. But, what about the expression 2*x*^2 + 2*x*? If, again, we substitute 1 for the variable *x*, we still get 4. How does this happen?

What we really did was simplify the original expression by distributing the 2 into the part in parentheses. So, we really haven't changed the expression at all - all we've done is rewrite it in a different form.

Because these two expressions are really the same, no matter what number we substitute for *x*, the results will always be identical. If we use 0, both expressions come out to 0. If we use 10, both expressions come out to 220. If we use 100, both expressions come out to 20,200. We get the same result no matter how large or small the number we use for *x*.

Some expressions will yield the same result for certain numbers, but not all. For example, let's consider the following expressions:

- 3
*x*+ 1 - 2
*x*+ 2

If we substitute 1 for the variable *x*, both expressions equal 4. But, what if we change that number to, let's say 0, the first expression then equals 1, while the second equals 2. Why do the expressions give the same answer for one value and not another? They're not equivalent expressions!

Remember that we previously discovered that equivalent expressions are really the same even though they look different. One is just simplified to give the second. That's not the case here, though. The two expressions above are completely different. It's just a chance occurrence that they yield the same solution when we substitute 1 for *x*.

In fact, if we graph the two expressions, we can see that they only intersect at that one point where they happen to yield identical solutions. However, they have their own tracks before and after that point because they're not equivalent expressions. While we're at it, let's see what happens when we graph the following equivalent expressions:

*x*+*x*+ 2- 2
*x*+ 2

Why is there only one line? Well, since equivalent expressions produce identical solutions for all values, their graphs are exactly the same. If we wanted to, we could graph a hundred equivalent expressions, and the result would still be one line because all the expressions would produce the same solutions.

So, if you're ever in doubt about whether two expressions are really equivalent, try plugging in several different values for the expressions' variables. If the expressions are indeed equivalent, they'll yield identical solutions for all of the values.

Let's review. Although two expressions may look different, they may really be the same expression written in different ways. For example, 2*x* - *x* + *y* and *x* + *y* may look different, but all we've done is simplify the 2*x* - *x*' part of the expression to *x*. So the two expressions really say the same thing even though they appear different. Therefore, they're **equivalent expressions**, and no matter what values we substitute for *x* and *y*, the expressions will always yield identical answers!

To unlock this lesson you must be a Study.com Member.

Create your account

Already a member? Log In

BackDid you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
14 in chapter 14 of the course:

Back To Course

High School Algebra II: Homework Help Resource26 chapters | 280 lessons | 1 flashcard set

- Using Tables and Graphs in the Real World 5:50
- Scatterplots and Line Graphs: Definitions and Uses 7:17
- What is a Parabola? 4:36
- Parabolas in Standard, Intercept, and Vertex Form 6:15
- Multiplying Binomials Using FOIL and the Area Method 7:26
- Multiplying Binomials Using FOIL & the Area Method: Practice Problems 5:46
- How to Factor Quadratic Equations: FOIL in Reverse 8:50
- Factoring Quadratic Equations: Polynomial Problems with a Non-1 Leading Coefficient 7:35
- Solving Quadratic Trinomials by Factoring 7:53
- How to Complete the Square 8:43
- Completing the Square Practice Problems 7:31
- How to Solve a Quadratic Equation by Factoring 7:53
- Parabolic Path: Definition & Projectiles 7:25
- Writing Equivalent Expressions: Definition & Examples 4:42
- Go to Algebra II Homework Help: Quadratic Equations

- DSST Information Guide
- GACE Early Childhood Special Education General Curriculum: Practice & Study Guide
- TExMAT Master Mathematics Teacher EC-4 (087): Study Guide & Practice
- ILTS Gifted Education: Practice & Study Guide
- Academic Writing Essentials
- Programming Basics in C++
- C++ Programming Functions
- Required Assignments for Criminal Justice 381
- Studying for Art 103
- Student Grouping Strategies
- CTEL Test Score Information
- CTEL Test Accommodations
- CTEL Test Retake Policy
- CSET Test Day Preparation
- How to Study for the VCLA Test
- Can You Use a Calculator on the CBEST?
- CTEL Registration Information

- Common Adverbial Clauses & the Subjunctive in Spanish
- Prison Reform: History, Issues & Movement
- Writing a Play: Script Format, Steps & Tips
- How Are ELL Students Identified?
- Helping Employees Identify Personal & Organizational Challenges
- Using Protagonists in Visual Media to Tell a Story
- Encapsulation C++ Programming: Definition & Example
- Program Memory in C++ Programming
- Quiz & Worksheet - Assessing the Cultural Background of ELL Students
- Quiz & Worksheet - Manorialism
- Quiz & Worksheet - Chromic Acid Test Reaction
- Quiz & Worksheet - West Egg in The Great Gatsby
- Quiz & Worksheet - Identifying & Analyzing Text Structure
- Flashcards - Introduction to Research Methods in Psychology
- Flashcards - Clinical Assessment in Psychology

- SAT Subject Test Biology: Practice and Study Guide
- High School Biology: Help and Review
- Organizational Behavior Textbook
- Julius Caesar: Help & Review
- Technical Writing: Skills Development & Training
- Praxis Mathematics: Rational and Irrational Numbers
- CSET English: Overview of American Literature
- Quiz & Worksheet - Prints in Northern Europe
- Quiz & Worksheet - Collage & Assemblage Methods & Works
- Quiz & Worksheet - European Battles for North America
- Quiz & Worksheet - Byzantine Art & Architecture
- Quiz & Worksheet - Money Market, Supply & Demand

- The Migration of Rococo to the United States
- Shakespeare's The Comedy of Errors: Summary & Analysis
- How to Use the GED RLA Prep Course
- Vietnam War During the Nixon Years: Learning Objectives & Activities
- National History Day Projects
- What are the NYS Regents Exams Locations?
- Causes of World War I Lesson Plan
- Best Psychology Books for Undergraduates
- Creative Writing Exercises for Kids
- AP Exam Registration Form
- Nervous System Experiments for Kids
- Creative Writing Competitions for High School Students

Browse by subject