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High School Algebra II: Homework Help Resource26 chapters | 281 lessons | 2 flashcard sets

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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!

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High School Algebra II: Homework Help Resource26 chapters | 281 lessons | 2 flashcard sets

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