# Writing Equivalent Expressions: Definition & Examples

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• 0:01 Algebraic Expresion
• 1:07 Equivalent Expressions
• 2:17 Beware Non Equivalence
• 4:02 Lesson Summary

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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.

## Algebraic Expressions

An algebraic expression is a string of numbers, variables, mathematical operations, and possibly exponents. For example, 4x + 3 is a basic algebraic expression. Or we could get a little more complex with 3x(2x^2 + 2x - 5) + 6y. 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: 2x + 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.

## Equivalent Expressions

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 2x^2 + 2x? 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.

### Beware Non Equivalence

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

• 3x + 1
• 2x + 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!

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