Two expressions are said to be equivalent if they produce the same number for any possible value of the variable. We'll bet you want to see some examples of this stuff in action. Don't ask how we knew. We had a feeling.
The expressions x and x + 0 are equivalent, since for any value of x, x and x + 0 are the same thing. Sorry, zero, but you're pretty worthless.
One way of expressing the fact that these two expressions are equivalent is to write an equation: x = x + 0. Any time you have an equation, the two expressions on either side of the equal sign are equivalent. That's kinda the equal sign's whole deal. It doesn't appreciate it when you write something like 1 + 1 = 3. Drives it batty.
The expressions x2 and (-x)2 are equivalent, since for any value of x we'll get the same value out of either expression. If x is 2, we find that 22 = 4 from the first expression and (-2)2 = 4 from the second expression. If x is 34,792, we get...oh, well, you probably got it from the first example. No need to wear ourselves out.
The goal is to create a toolbox of ways to modify an expression to an equivalent expression. We're not, of course, talking about a literal toolbox. You'll never receive an emergency call to hop over and fix an overflowing toilet with your trusty ratcheting box variable. Instead, these moves are the rules we have for working with algebraic expressions. Once we finish developing this (virtual) toolbox, we'll use our tools to solve equations and more. Okay, maybe the occasional leaky faucet, but that's where we draw the line.
There are several different things we can do to rearrange expressions. We've divided them into the following list of allowable moves:
5. Combining like terms
6. Getting rid of parentheses
Use any moves that aren't in the "toolbox," and you might draw a penalty flag. Incorrect Rearrangement of Expressions: 15 yards and loss of down.
Using combos of those moves together to rearrange an expression into an equivalent expression is called simplification.