Algebra is the science of equations. That is, it tells us how to work statements that tells us when something is equal to something else. This is tremendously convenient, especially when you are trying to figure out an unknown quantity. With algebra, you can use a known quantity to get to the unknown quantity.
There are four basic equations in algebra, corresponding to the four operations that you do in arithmetic. Those operations are: addition, subtraction, multiplication, and division. The four basic equations involve addition, subtraction, multiplication, and division.
The way to solve them is to do the opposite of the indicated operation. That is, in an addition equation, you subtract, in a subtraction equation, you add, in a multiplication equation, you divide, and in a division equation you multiply.
This is called the Law of Inverse Operations.
It’s a simple concept to grasp. If you have $10 and I gave you $10 more, you would have $20. If I were to take that money back, you’d be back to $10. So, if I wanted to figure out how much money you had to begin with, I’d set up the following equation:
To figure out what A, the original amount, was, I’d subtract $10 from both sides of the equation and I would be left with $10.
Similarly, if someone stole $10 from you, leaving you with $10, I could restore your wealth by adding $10 back.
Multiplication works the same way.
Suppose the eight of us were to invest $8 each. A math wiz would multiply each of our holdings by $8 and come up with a total investment of $64.
If we wanted to know the original amount, we could divide the $64 by 8 and come up with it.
Similarly, if we were to divide $64 by $8 and get $8, but wanted to reverse that, we would multiply by $8.
To solve an equation, we reverse the operation that is indicated by the equation.
To solve an equation, you do to the one side the same as to the other.
You will not be able to solve anything if you only work one side. If X +4 =16, you will never, ever learn what X was by simply subtracting the 4 from the left side. You have to do it to the right to get the answer.
Equations are a form of balance. If you and your friend are on a teeter-totter and President Trump sits on your friend’s side, you will go flying. Unless you have Chris Christy, former governor of New Jersey, sit on your side. In learning algebra, always keep the teeter-tooter in mind.
Now that we got that straight, what do we do when an equation requires more than two operations?
We do them in steps. Do the steps have any order? Yes, they do. Remember PEMDAS? (Please Excuse My Dear Aunt Sally?) In other words, we do parentheses, exponents, multiplication/division, then addition/subtraction when doing multiple operations.
In solving an equation, we reverse that. Why? Because we want to get back to the original point. So in the following two-step equation:
2X +4 =16
We first reverse the four. To reverse addition, we do subtraction. So we subtract four from both sides.
Now we reverse the multiplication. We multiplied by 2, so we divide both sides by 2.
What could be simpler? Well, actually, one-step equations which we discussed at the beginning. Can it get more complicated? Oh, yes.
We’ll deal with that later.
Dr. Fred Young