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How do we know whether two expressions are "equivalent?" This might seem like a relatively easy process, but things can get a little tricky when we start working with expressions that have variables. Fortunately, there are a number of tricks we can use to determine equivalent expressions -- even with variables.

Expressions are equivalent when they mean the same thing. In other words, equivalent expressions have the same reduced or simplified forms.

Consider these two expressions:

${3}^{2}\mathrm{+}1$ and $5\times 2$

Both of the expressions are equal to 10. Therefore, they are equivalent.

But what happens when we add variables into the mix? How do we determine whether expressions with variables are equivalent?

Let's start with the expression $5x+2$

First, let's rewrite it in a very complicated way:

$x+x+x+x+x+1+1$

We can gather the variables together in any way that's convenient. For example, we could write this equation as:

$2x+3x+1+1$

Or:

$x+4x+2$

These are all equivalent expressions because they mean exactly the same thing. It doesn't matter what we substitute for $x$ -- we always get the same result.

In other words, expressions are equivalent if their value is the same -- regardless of variables.

Consider these two expressions:

$2y+5y-5+8$ and $7y+3$

Are these two expressions equivalent?

First, we'll need to combine the like terms of the first expression:

$2y+5y=7y$

Now let's combine the other like terms:

$(-5)+8=3$

Putting this together, we get:

$7y+3$

This is exactly the same as the other expression, which means that they are equivalent.

Let's consider two more expressions:

$6(2a+b)$ and $12a+6b$

Are these two expressions equivalent?

Let's start by using the distributive property to expand the first expression:

$6(2a+b)=6\times 2a+6\times b$

$=12a+6b$

This is exactly the same as the second expression. Therefore, the two expressions are indeed equivalent.

Let's consider another pair of expressions:

$2x+3y$ and $2y+3x$

In this situation, we can run a few tests by substituting values for x and y:

Let's assume that $x=0$ and $y=1$ .

$2\left(0\right)+3\left(1\right)=0+3=3$

$2\left(1\right)+3\left(0\right)=2+0=2$

Remember, equivalent expressions should be the same regardless of
the value of their variables. Because we have at least one situation
in which these expressions are not the same, they are
*not* equivalent.

Consider one final pair of expressions:

$\frac{(3\times m\times m)}{m}$ and $m+m+m$

Are these two expressions equivalent?

We can start by canceling the common terms. This leaves us with a new first expression:

$3m$

Now let's combine the like terms of the second expression:

$m+m+m=3m$

Now we know that the two expressions are equivalent. But wait --
what happens if m = zero? If this is the case, the first expression
is not defined. So we know that there is at least
*one* situation in which these two terms cannot be equivalent.
This means that although these terms *seem* equivalent, they do
*not *pass the test of equivalency.

Equivalent expressions must have equivalent domains.

Greatest Common Factors (GCFs)

College Algebra Diagnostic Tests

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