3.2 SAT Math - Like Terms

With more complicated equations, it may be necessary to first combine like terms.  Like terms are terms that contain the same variable, raised to the same power. 

So 2x, 3x, and ½x are all like terms, but 2, 3x, x², and ½y are not.

-Trick to Watch Out For on the SAT math section: A number that is attached to a variable (such as 3x or x/2) can NEVER be added or subtracted to a regular number.  These are unlike terms are cannot be combined.  So 4y + 2y can be combined into 6y, but 4y + 2 CANNOT be simplified further.  Likewise, variables with exponents represent a DIFFERENT number than the variable alone.  So 3x +x² is NOT 3x²!  It is at its simplest as 3x +x² unless you choose to factor (an option that will be covered later.)

-Hint: Many difficult-looking problems are simply testing your ability to combine like terms.  Get all your variables together on one side, all your numbers on the other, and you may find you have a much easier problem on your hands than you thought:

Pick one side to clean up first, then do the other, then do the opposite to both sides to get all your variables on one side and your numbers on the other:
                       

Solve for the Variable

Solving in terms of a variable is done the same way; you just won't get a single number at the end:

Solve for y in terms of x. 3x + y = 42

Get the variable you want on one side, and everything else on the other:

3x + y = 42
-3x         -3x
y = 42 - 3x

Keep in mind that, in this example, the above answer is the simplest you can get without factoring.  42 is a number, 3x contains a variable, therefore they are not like terms.

-Trick to Watch Out For: Be sure to isolate the correct variable!  The testmakers like to make it look pretty obvious to solve for one variable, but are in fact asking about the harder one – but because isolating a variable is not considered very hard math, it might still be a lower-numbered problem than you would expect, and you might make the mistake of speeding through.  Example:
                                
It may seem easier or more obvious to solve for y here, or even x, and variations on those answers are options…but answer choice (C) should be the first one you eliminate.  If they’re asking you to solve for z in terms of x and y, z should not appear in the correct answer.  (The correct answer is (E) .)

If you didn’t get the correct answer, if you’re staring at your screen right now flummoxed and flabbergasted because you have no idea what just happened to get from that ugly equation to that ugly answer, don’t worry.  A solution is on its way.

But first, a little more review: Next - Subsitutions