The branch of math that we call algebra deals with relationships. The key to algebra however is not memorizing equations or processes, or even rote repetition, it is in understanding one simple concept, the equals sign. If you do, then you will be able to solve the most complex problems, but if you don’t understand the true meaning of =, then you are in for a tough time.
Now what does = mean? It seems like a simple enough concept. Most of us would probably agree that = means that two things are the same. That’s almost correct. = means that both values on each side of the = sign are the same value. That is not the same as saying that they are the same thing. Let’s look at this:
1 dollar bill=1 dollar bill
This is a correct statement because the value of each dollar bill is the same. It also happens that they are the same thing. Now try this one:
4 quarters=1 dollar bill
These are NOT the same thing, but they have the same value. This is an important distinction because in algebra you will be solving equations in which there is an unknown value, usually denoted by an “x,” (but other letters are sometimes used too) and your job is to find what x is equal to. Both sides of the equation will rarely look the same, but they will be the same value. So if we have:
x + 2 = 5
That means that if we take some number and add 2, we get 5. Of course this is easy to do in your head, but what you did without realizing it is this:
x + 2 = 5
x + 0 = 3, or just x=3
You subtracted 2 from both sides, which is easy to understand, but when it gets more complicated, it gets harder to know what to do. You already know what = means, but keeping it = is what you have to do in algebra. Let me explain what I mean by using an analogy:
You have a scale and your job is to keep it balanced. This of course is done when you have the same amount of weight on both sides. Here’s our scale with some weight on it:
Notice that the weight is the same on both sides, so although the weights are not the same, they have the same value, 10. Now remember, we have to keep this thing balanced, so if something is done to one side of the scale, the exact same thing must be done to the other. So let’s say that you take away 2 from the left side; you get:
Of course now it isn’t balanced anymore, and I suspect that you realize what we have to do. That’s right, we have to take away 2 from the other side. So let’s do that, and we get a balanced scale again:
That wasn’t so hard, but it can get a little trickier when you are doing multiplication and division. You just have to remember though, that we are making changes to both sides, not just individual numbers. So when we took away 2 from the left and right sides, it was from the whole side, not just one part of the side. Now let’s do a division problem so you can see what I’m talking about:
First, let’s take our original scale:
Now this is important for you to understand. What we will do is divide the 10 by 2. That gives us 5, right? So now we have to do the same thing to the other side. We divide the whole other side by 2, not just the 4, 3, 2, or 1, but the 4 and the 3 and the 2 and the 1. So now the scale looks like this:
Notice how each number on the left side was divided by 2, just like every number on the right side was divided by 2 (even though there was only one number). The way we could right this in an equation is:
4 + 3 + 2 + 1 = 10
4 + 3 + 2 + 1 = 10, and that’s the same as:
4/2 + 3/2 + 2/2 +1/2 = 10/2, which is the same as:
2 + 1.5 + 1 + 0.5 = 5
In algebra, we tend to write like this:
1/2(4 + 3 + 2 + 1) = 1/2(10), but don’t worry, its the same. The parenthesis just mean that we are multiplying the ½ by everything inside.
Make sense? Now let’s try some real algebra problems, but remember, the exact same concept applies. If we change one side of an equation, we have to do the exact same thing to the other side and we need to make changes that will get x by itself on the left side. So let’s look at this problem:
3x + 7 = 13
First let’s get rid of the 7 on the left side, but remember, we have to do it to both sides:
3x + 7 = 13
3x = 6
So now we know that 3x = 6, so let’s get rid of the 3 by dividing both sides by 3 or multiplying by 1/3, they’re both the same (and then we’ll only have one x by itself on the left side).
1/3(3x) = 1/3(6)
x = 2
Now we can check and see if we are correct by plugging in a 2 for x in the original equation:
3(2) + 7 =13. Sure enough, we got it!
Now let’s try one that is a little tougher. Just remember, we do it to one side, we do it to the other!:
3x + 2 = 2x + 4 (the dots don’t mean anything, they just make sure the 2 is in the right place)
Don’t be scared, you will learn as you practice which operations to do in what order to solve it the quickest, but at the end of the day it doesn’t matter. You can do anything, as long as you do it to both sides. Now it may not get you closer to the answer, but it will still be a true statement. So let’s tackle this sucker!
First of all, let me rewrite the equation in a way that may help you understand better:
3x + 2 = 2x + 4
You don’t have to rewrite it this way, but it may help you to see that we need to first multiply the left side by 2 to get rid of the fraction. And of course you know you will have to do the same to the other side:
2 (3x + 2) = 2(2x + 4), so we get
3x + 2 = 4x + 8.
Now let’s get rid of the 2 on the left side:
3x + 2 = 4x + 8
3x = 4x + 6
Now let’s get the 4x together with the 3x by subtracting 4x from both sides:
3x = 4x + 6
-x = 6
Now we just have to multiply (or divide) by -1 to get x by itself, and we’ll have our answer!:
So x = -6. If you want to plug -6 into the original equation, you will find that it works (of course).
That’s how you do it. You are just trying to get x by itself, but to do so, you must always do the same thing to both sides of the equation. There are lots of practice problems with solutions if you need some practice. Even if you understand fully, a little practice is still important.