Algebra Explained in 5 Minutes
Consider this problem, "what number, when added to 5, gives the result 21".
Instead of a sentence, this problem can be written much shorter and clearer as an equation, like this..
where x denotes the number we are trying to find.
Of course, we could also write it as x+5=21 and this is exactly the same equation. Or we could write 21=x+5 which is of course the same thing.
If we manage to find x we say that we've "solved" the equation. Can we solve this equation? Well, we could guess a few numbers for x and try them out. Does x=9 work? Let's see, 5+9=14, so x=9 is not a solution. After a few tries we get the solution, which is x=16.
Guessing a solution is perfectly fine, but it's very time consuming, especially for more complex equations. Of course, we could program a high speed computer to guess solutions and try them out ultra fast until we finally hit on the right solution. And for some very tough equations this is indeed the method used. But this method has a huge flaw.. if it fails to find a solution it does not mean the equation has no solution. That's because even the fastest computer can only make a limited number of tries.. and the actual solution may be something we never get around to trying.
So, coming back to our equation 5+x=21 we should ask if there is a foolproof method that's guaranteed to find the solution. The answer is yes, and it's all about the = sign. Once you truly understand this simple sign solving the equation is easy.
So what does this sign really mean? It means the "object" on the left of the sign is the same exact object as that on the right. They are the same thing.. exactly the same thing. They are the same exact mathematical object but just written in different ways. So there's really only one object!
OK, so our equation says that 5+x is exactly the same object as 21. So, if I do something to 5+x and then I do the same thing to 21 the results will still be equal. Cool. So lets subtract 5 from 5+x to get the result x. Now do the same exact thing to the other side, I'll subtract 5 from 21 to get the result 16. But these two results must be the same, so I can write them as equal to each other, that is x=16.
Bingo, we've solved the equation without any guessing!
Also, I'm not sure if you noticed this, but we just did some basic Algebra. Don't let Algebra intimidate you, it's just the art of manipulating equations until you get what you want!
Let's look at a slightly more complicated example..
To solve it we want to isolate x on one side and get all the other stuff over to the other side. Here's a method I use. It's exactly the same technique as above, but it's faster and easier to handle. Or at least I think so, and I've used it over the years to do massive amounts of algebra!
First move the 2 over to the other side. It was adding, so when it moves over it subtracts, like this..
Now move the 3 over. It was multiplying, so when it moves over it divides, like this..
This technique is quite general and can be used for any equation. But notice that the order in which you do things is important. For example, you need to get the 2 over to the other side before you can handle the 3.
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Content written and posted by Ken Abbott firstname.lastname@example.org