A Third Math Lesson: Order of Operations

Order of Operations

Or, “Manipulating Equations to Get What You Want on Your Own TERMS” (Get it?  It’s a pun.)

You’ll notice that in both of the Isolating the Variable exercises, we dealt with addition and subtraction first, and then dealt with multiplication and division last.  In Math, this is called the proper Order of Operations.  Frankly, it’s not just the proper order, it’s the easiest order.  Another way of thinking about this is to say that we deal with an equation a term at a time. 

An equation is made up of terms.  What’s a term?  Well, a term is a chunk of an expression.  So let’s look at our old friend:

4x + 7y  - z = 42 

That’s an equation, made up of the expression 4x +7y – z, an equal sign, and the number 42.  In the expression 4x + 7y – z, we have three terms:  4x, 7y and –z.  We could also call it -1z, but in Math, we usually leave off the 1 in a term, just to keep it simple.  One z = z =1z.  Z.  We’ll leave it at that.

A term is not any old chunk of the expression.  For example, we can’t call 7y-z  a “term.”  That, my friend, would be called an “expression.”  Just a smaller expression than the one above.  A term is a nice, tidy chunk, all stuck together, without the glue of a “+” or a” – “ holding it together.  A term is just held together by multiplication or division.  So these things are all terms:

8x        17b      -2m³        149y²       -83x       ¼ h

Why do we care what a term is?  Well, because the first step in the process of Isolating the Variable is to isolate the term that contains the variable we want to isolate.  Let’s look at an example:

2a – 14b + 72c = 82

Let’s isolate the variable b.  In order to do that, we’ll need to start by isolating the term that contains b, or “-4b.”  We’re  gonna start by kicking all the other terms out of the room.  By that I mean you, 2a, and you, 72c.  Pack up your stuff, cuz you’re heading over to the other room.  But we can’t just put them there, because our equation won’t be true anymore.  So let’s do it right, and use math:

You first, 72c.  Get ready, cuz here you go:

2a – 14b + 72c = 82

2a – 14b +72c -72c = 82 -72c

2a – 14b + (72c -72c) = 82 -72c

2a -14b + 0 = 82 -72c

2a – 14b = 82 -72c

And now you, 2a.  You’re next:

2a – 2a – 14b = 82 -72c -2a

(2a – 2a) – 14b = 82 – 72c -2a

0 – 14b  = 82 – 72c – 2a

-14b = 82 – 72c – 2a

Ta da!  Now we’ve isolated the term with b in it, so we’re ready for the last step.  Right now we can see what b looks like wearing a negative 14 coat (or a -14 coat).  Let’s get him naked!  How do we do that?  Well, he’s a b multiplied by a -14.  To undo that, we’re going to have to divide both sides by -14 (dividing is the opposite of multiplying).  So now we have:

-14b = 82 – 72c -2a

  -14            -14

b = 82 – 72c -2a

            -14

So a quick review on Order of Operations when isolating a variable:

1.       Isolate the term containing the variable you want to isolate (or solve for).  To do this, perform the opposite operation of what got those terms there in the first place.  Do the same operation to both sides of the equation.

2.      Perform the opposite operation to what has already been done to your variable.  Again, make sure you do the same operation to both sides of the equation.

3.      Bask in the glory that is your naked variable.  You have solved the equation for your chosen variable.      Do a little dance.