THE ESSENCE OF ALGEGRA
Algebra is your first step from arithmetic to higher mathematics. It is necessary to master the Essence of Algebra for you to proceed to trigonometry, analytic geometry, and calculus. Once I learned and understood the purpose of algebra, I found the reason for the procedures of Algebra. The procedures are easy to understand, but I found without understanding the purpose of Algebra I could not get a sense of the meaning of algebra.
Algebra can be understood by recognizing that algebra provides the procedures for manipulating equations and that algebra consists of nothing more than a bunch of procedures. You may not realize it, but you have used algebra intuitively. That is, if I were to say that John is 20 years old and John is 5 years older than Mary, then how old is Mary? You would simply reason that since John is 20 years old then 20-5 = 15, which is Mary's age. You were using algebra intuitively. If we assume John&'s age can be represented as X and Mary's age as Y, then the equation expressing Mary's age as a function of John's age can be expressed algebraically as:
Y = X - 5 and substituting 20 for X, which is John's age then,
Y = 20-5 = 15
Where: X and Y are called variables and X is called the independent variable since it has the known value (i.e., 20), and Y is referred to as the dependent variable and is a function of (dependent on) X. That is, Y depends on the value of X. The equal sign makes it an equation. Remember that, to maintain the equality, whatever you do to one side of an equation you must also do to the otherside.
One of the first things you will learn is the relationship of signs:
+*+ = + plus times plus = plus
+*- = - plus times minus = minus
-*- = + minus times minus = plus. and
-*+ = - Minus times plus = minus
That's all there is to know. Easy, right?
Now let's move a variable from one side of an equation to the other.
Here's an equation:
Y=X-5 let's move the -5 to the other side of the equation.
From above remember that what ever you do to one side of an equation you must also do to the other, so:
Add 5 to both sides of the equation:
Y+5=X-5+5=
Y+5=X
Notice that when the original -5 moved to the other side of the equation the sign changed. That's all you need to know. You need not go through the procedure of adding to or subtracting from both sides of the equation you simply change the sign of the item that you are moving to the other side of the equal sign.
Notice that when we were discussing signs it was simply a procedure and when we moved a variable from one side of an equation to the other if was simply a procedure. It is procedures such as the foregoing that you will learn as you cover the rest of this text.
As development of this page progresses, the study of Algebra will be extended here to cover a sufficient number of procedures so that you will have no problem going on to trigonometry and calculus.
