## The Essence of Algebra

There are two basic features of algebra that are very very important to the person studying algebra, variables and equations. This may seem obvious. But, for the student who is struggling with algebra, a better understanding of these two ideas will probably help, a lot.

Variables

In algebra, we use letters. Different letters are sometimes used for different purposes. But basically, a letter represents a number, and we don't know its value. Often, we want to find out its value. How about this expression?

n-n+5

We don't need to know what n is. That expression is equal to 5, no matter what n is. So, sometimes we don't find the value of a variable. We can just cross out the n-n and leave the 5, because n-n+5 IS 5. n-n+5 is a number, and it is the same exact number as 5.

In algebra, we have expressions involving variables and various arithmetic operations. These expressions are also numbers. Are there any exceptions to this? Yes. In algebra, you can never divide by zero. See my article Two Equals One?

Other than division by zero, there is one thing that we can usually do with an expression like this, we can substitute another form of the same number. That is what we did, when we substituted 5 for n-n+5. They are just different forms of the same number.

Above, I said that different letters are sometimes used for different purposes. How does that complicate things? Mainly, it clarifies things. x, y, z are generic variables, and we probably want to solve for them (figure out what their values are), or graph them. a, b, c... often take the place of constants, when we don't care what their actual values are. We may be saying something about ax+b, in general, where a and b are constants and x is a variable. n and m might have some specific meaning, like maybe they are integers.

Equations

An equation is something of this form:

number1=number2

We often have algebraic expressions (involving numbers and variables and arithmetic operations) on either side (or both sides) of an equation. An example is:

2x=3

The numbers on both sides (2x and 3) are the same number. That's why we have the equal sign. It's as if we have a scale for measuring weights. On one side, we have a weight that is 2x pounds. On the other side, we have 3 pounds. And the whole thing balances.

What can we do with an equation? There are mainly two things to do with an equation:

1. Substitute any number for a different form of the same number
2. Do the same operation to both sides of the equation

What can we do to both sides of an equation? Actually, we can do almost anything, except divide by zero. Think of the balance, again. We can add the same weight to both sides. We can subtract the same weight. We can multiply the same number (for example, we can double the weights on both sides, since they are the same weight). And we can divide the same number, as long as we are careful not to divide by zero. We can do stranger things, like take square roots, logarithms, sines, powers... Remember, they are the same number, on both sides. So we can do the same thing to both of them. Also, thinking of the equation as a balance may help.

So, let's say that you are factoring quadratic equations. You are now into tough concepts. You've left the ideas of variables and equations far behind, haven't you? No.

You may have learned a few other things, some tricks and ideas. But you're mainly just substituting and doing the same thing to both sides of an equation. And, you will be doing those things in Calculus. So, try to understand these ideas.