Algebra Online Learning Center

INTRODUCTION

There is a person inside you who always says," You can't do that. Leave it." It is not true, but you believe him because you get some benefit from that. You get a good excuse to slip away from your work. You can go to your friend, watch TV, whatever you want to. However you can not feel genuine pleasure from your entertainment because there is the other person inside you who wants to be proud of your achievement. He tells you'll keep trying! You can do that! You are smart." Believe him! This book will show you the way of thinking that help you to understand math. This book is for everyone who wants to feel free with a mathematical formula, who wants to comprehend the beauty of algebraic expressions. Did you feel frustration looking at your math textbook? Forget it! Fall in love with math!

1. HOW TO SOLVE THE ALGEBRAIC EQUATIONS

Math is a very pleasing field of activity. Having just a pen and a piece of paper you can invent whatever you want. You can wander around the paper with numbers and symbols caring about just one thing: equality should be equality, nothing more. Let us imagine that you are the first great mathematician. All people know only arithmetic: how to add, how to subtract, how to multiply and how to divide. In school they study boring things like that:

2 + 3 = 5 or 7 - 4 = 3

You are the first who suspect there is a way to express a common idea of the written above algebraic equations.

Once you will write: a + b = c or c - a = b. Now you discovered common rules that help people to solve any algebraic equation. To make the discovery you have to perform experiments with numbers. Let us write simple algebraic equation:

4 + 8 = 12

Let us add any number to the left side of the algebraic equation.

4 + 8 + 3 = 12

What did you get?

15 = 12

You are wrong! How can you fix your algebraic equation? Right! You must add the same number to the right side of the algebraic equation. 4 + 8 + 3 = 12 + 3. What did you get? 15 = 15

You discovered the first rule for algebraic equations. This rule says: "If you add the same number to the left side and to the right side of an algebraic equation, this algebraic equation still will be true." To express this rule in a common way you can write:

If a + b = c then a + b + n = C + n where a, b, c, n - any numbers. Are you not a genius? Yes, you are. Go ahead. Let us try to make the next experiment. What happens if you subtract any number from the left side of an algebraic equation?

5+2=7 5+2-5=7

What did you get? 2 = 7

You are wrong, but you know how to fix your algebraic equation. You must subtract the same number from the right side of the algebraic equation.

5 + 2 - 5 = 7 - 5. Then 2 = 2.

Congratulation! You discovered the second rule for algebraic equations. This rule says:' If you subtract the same numbers from the left and the right side of an algebraic equation this algebraic equation still will be true."

Or you can write:

if a + b = c then a + b - n = c - n where a, b, c, n- any numbers.

What kind of experiments can you do else? You can multiply one side of an algebraic equation by some number. Let's write an algebraic equation:

5 - I = 4

What happens, if you multiply the left side of the algebraic equation by 7.

(5 - 1)7 = 4 then 28 = 4

It is not true. Try to multiply both sides of the algebraic equation by 7. (5 -1)7= 4 x 7. Then 28 = 28. You discovered one more rule for algebraic equations. The third rule says:" If you multiply the left and the right side of an algebraic equation by the same number, this algebraic equation still will be true."

You discovered four rules (ways) of solving an algebraic equation:

1. Add the same number to both sides of an algebraic equation.
2. Subtract the same number from both sides of an algebraic equation.
3. Multiply both sides of an algebraic equation by the same number.
4. Divide both sides of an algebraic equation by the same number.

if a - b = c then (a - b)n = cn

One more question. What happens if you divide one half of an algebraic equation by any number? 4 + 6 = 10 (4 + 6): 2 = 10 then 5 = 10.

"How many times will you make the same mistake?" you can ask yourself and you are right. You must divide both sides of the algebraic equation by the same number.

(4 + 6): 2 = 10: 2 then 5 = 5 You discovered the fourth rule for algebraic equations. This rule says:" If you divide the left and the right side of an algebraic equation by the same number the algebraic equation still will be true." So, you can write:

if a + b = c then (a + b): n = c : n

Where a, b, c, - any numbers, but n =/= 0 because you can't divide numbers by 0. "What kind of profit can you get from these rules?" People will ask you. "Well, you tell them." You can use these rules to solve any algebraic equation." Let us write an algebraic equation where one number is unknown.

X - 3 = 11 How we Can Solve this algebraic equation? Let us try to apply the first rule: If you add to the left and the right side of an algebraic equation the same number this algebraic equation still will be true.

For our algebraic equation it is convenient to add 3 to both sides of the algebraic equation.

x - 3 + 3 = 11 + 3. Since, -3 + 3 = 0. Then X = 11 + 3 = 14

Let us try to solve the algebraic equation where all numbers are represented by letters.

X - b = c

Apply the first rule to solve this algebraic equation X - b + b = c + b

Since, -b + b = 0, then X = c + b.

To solve the algebraic equation X + b = c we can apply the second rule.

If X + b = c then X + b - b = c - b X = c - b.

The next sample: X + 7 = 15 then X + 7 - 7 = 15 - 7 and X = 8

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