I’m Breaking my Oath as a Mathematician
I’m going to do it! I’m going to break the hallowed math teacher’s oath and-like the rogue magicians-pull aside the veil and show you the secret to solving equations. Any equation. All equations.
This insight is rarely gained by the initiated since, by its very nature, math must be learned in pieces and then synthesized into a holistic understanding. Therefore, very few students are able to look across the entire K-16 math spectrum and see the golden thread that runs through all of the math courses that are arithmetic and algebra based.
But, in my own humble way (ahem), I’m going to show you that one common approach that helps you make sense out of solving equations.
Have I built this up enough yet? Do I have your attention so we can have that teachable moment? I hope so because here it comes (in caps, no less):
WE SOLVE EQUATIONS BY USING OPPOSITE OPERATIONS TO “UNDO”
That’s it! Keep in mind that some of the most profound insights at first seem simple, and maybe even obvious. So let’s unpack this, shall we?
An Arithmetic Example
First, let me illustrate by using arithmetic. Imagine I walk over to where you are sitting and I give you this thingamajig represented by the X:
Me ———————————–X You
This is equivalent to ADDING to you.
Now, if I want to “undo” what I just did, I would take it away and leave you empty handed:
Me X ———————————–You
This “taking away” is the same as subtracting.
Therefore, since subtraction “undoes” what addition did, subtraction and addition are paired opposite operations.
An Algebra Example
Let’s apply this reasoning now to algebra by starting with the following simple equation along with my comments:
X + 3 = 5 Focus on the X, that is the prize or the target-the thing we really want.
So here is X, minding its own business, and someone came along and added 3 to it.
We need to UNDO what was done.
Since something was added, we need to use the OPPOSITE OPERATION to UNDO
X + 3 – 3 = 5 – 3 Subtracting three UNDOES what was done and WHATEVER WE DO TO ONE SIDE OF AN EQUATION
WE MUST DO TO THE OTHER SIDE.
X = 2 We are left with the X by itself on one side and the 2 on the other. So X must equal 2.
A More Complicated Algebra Problem
2X – 6 = 14 We’re going to use the same reasoning with this problem but adjust how we thing about it a little bit
Imagine that the X that we want is a Birthday present. Have you ever had someone tease you by wrapping the present and then
putting it in a box and wrapping it with new paper, etc.? When you begin the process of getting at your present, you have to take off the
outside paper first!
In this example, the 2 is paper that is close to your present and the 6 is paper that is farther away, on the outside. So the outside
paper (the 6) has to go first.
2X – 6 = 14 Since WE SOLVE EQUATIONS BY USING OPPOSITE OPERATIONS TO UNDO, we look at the 6 and see that it was subtracted.
Therefore, we must use the OPPOSITE OPERATION of addition to UNDO.
2X -6 + 6 = 14 + 6 Whatever we do to one side, we must do to the other. But keep in mind that our focus is on the present (the X).
If you wanted to, you could even draw a line through the – 6 and the + 6 to show how adding 6 UNDID the subtracted 6.
2X = 20 Now you’ve unwrapped the “outer paper” and you’re ready to get rid of the “paper” that is closest to the present.
Since the 2 is multiplied times the X, you must use the OPPOSITE OPERATION to UNDO. You must divide both sides by 2 to UNDO.
2X = 20 Dividing by 2 UNDOES and WHATEVER WE DO TO ONE SIDE OF AN EQUATION WE
2 2 MUST DO TO THE OTHER SIDE.
X = 10 Since we are UNDOING, the two 2s on the left side cancel each other leaving us with X on the left side and 10 on the right and the
equation is solved.
ex = 5 This is a more advanced equation and depends upon you knowing that logarithmic and exponential function are, in fact, OPPOSITE
to each other. Therefore, since what we see to the left is an exponential (see the little x raised into the exponent position?), we need
to use the OPPOSITE to UNDO and apply a logarithm to both sides.
ln ex = ln 5 Since I see the symbol e (the mathematical constant that is approximately 2.71828) I’ll use the natural logarithm (ln) which is base e
lne ex = ln 5 This is getting extremely hard to see in small type, but I know that when those two e’s are in place like we see to the left, that the
entire left side of the equation becomes X and the right side becomes 1.609 (ln 5).
So I recognized an exponential equation and used the OPPOSITE to UNDO and solve.
It Just Goes On and On
This truly is the golden thread that runs through equation solving. For example, differential calculus and integral calculus involve opposites. It all involves unwrapping the extras to get at the prize.
Keep in mind that this is a way of looking at equation solving from a holistic perspective. You still have to know quite a bit of the detail. But this holistic, overarching insight into algebra can help you pull the fragments together and make sense of how you can have a common approach to solving equations.