Introduction to Algebra


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Algebra has been around since well before 200 AD. It was then that an Alexandrian Greek named Diophantus first used letters and/or symbols to represent unknown numbers. Even before Diophantus, as early as 1800 BC, Babylonian mathematicians were solving complex linear and quadratic equations. Around 800 AD Abu Ja'far Muhammad ibn Musa al-Khwarizmi, who was from Baghdad, now a city in Iraq, is credited with giving us the word "Algebra." His books are considered by many to be the first to teach Algebra in an elementary form.

Maybe some of you have heard that algebra is hard. Guess what? It isn't! In fact, I'll bet you probably already know a little! Algebra replaces unknown numbers in a problem with letters. Let's look at an algebra problem that I'll bet most of you can already answer.

2+2=Z

Can you guess what Z is? If you guessed 4, you're absolutely right! See this isn't so hard after all! Did we have to use Z for the letter? No! We could have used W, E, Q, or anything else for that matter. All the letter does is represent a number that we don't have the answer for yet.

Have you ever thought about how a problem with the correct answer is balanced? What I mean by "balanced" is that both sides are equal to each other. Think about the problem above.

2+2=4

If you look closely, 4 is on both sides of the equals sign. 2+2 is 4, and of course 4 is 4. You could rewrite 2+2=4 this way.

4=4

If this sounds kind of weird, think about a problem that doesn't have the correct answer.

2+2=5

This problem isn't balanced, because it says 4=5. Of course we all know that isn't true! We always want our Algebra problems, or any other problems for that matter, to be "balanced" so that we will have the correct answer! Unbalanced problems are always going to be wrong!

If all algebra problems were like the first one, we really wouldn't need algebra at all. Algebra is effective for finding a number when we already know the answer, but may not know one of the other numbers it took to get the answer. When would that happen you say? Consider this simple problem. You have 2 books on your bed. You know that there are a total of 4 books in your room. How many other books are in your room besides the 2 on your bed? You probably already know the answer, but let's write the problem algebraically.

F+2=4

Make sure you see where we got the problem above. You have 2 books on your bed. That's where the 2 came from. You know that you have a total of 4 books in your room. That's where we got the 4. What we want to know is how many other books there are in your room. That's where we get the F. F represents what we don't know. How are we going to find the answer for F without guessing? We have to keep the problem balanced, so that both sides equal each other at all times in order to get the right answer. We need to get F on one side of the equals sign by itself in order to find out what F equals. Let's work some "magic." We know that 4=4. We also know that if we subtract 2 from both sides of the equals sign that 2=2, and the problem is still balanced.

4-2=4-2
(which is the same as)
2=2

What does this have to do with F+2=4? Let's try subtracting 2 from both sides and see.

F+2-2=4-2
(which is the same as)
F+0=2
(which is the same as)
F=2

We know that 2-2=0 on one side of the equals sign and that 4-2=2 on the other. We are left with F+0=2. Of course adding 0 to something doesn't add anything at all so really what we have is F=2. There's our answer! F=2. To check our answer we "plug" the 2 back in for F in the original problem of F+2=4. When we do we'll find that we get 2+2=4. As you have probably already realized, we have the correct answer for F!

With algebra problems, the easiest way to find an unknown number is to always get the letter on one side of the equals sign by itself. When you do that, the answer lies on the other side of the equals sign just like it did in the problem above. With addition algebra problems like F+2=4, you have to subtract the 2 from both sides to get F by itself. Why do we have to do it to both sides? We do it to both sides in order to keep the problem balanced. If we just subtracted 2 from one side, we would have an unbalanced problem and the wrong answer. Try it and see!

F+2-2=4
(which is the same as)
F+0=4
(which is the same as)
F=4

When you plug 4 back in for F to check it, you're going to find that 4+2=4 is WRONG! The reason is because we didn't keep the problem balanced! What you do to one side of the equals sign you MUST do the same exact thing to the other in order to keep the problem balanced!

Now let's consider a subtraction algebra problem. In order to work the addition algebra problem we had to use the opposite of addition, which is subtraction, to find the unknown number. With subtraction, the opposite is addition. This time we'll have to add a number to both sides of the equals sign in order to get the letter on one side by itself and keep the problem balanced. Here's the subtraction algebra problem.

B-3=2

Some of these problems are probably easy for many of you to guess without using algebra. That's cool, but we want to learn how to do things algebraically, so let's work them using algebra instead of just simply guessing to find the answer. In order to get B on one side of the equals sign by itself we need to get rid of the "-3" that we are subtracting from B. We can do that by adding 3, because we know that 3-3=0. We also know that we need to do this to both sides of the equals sign in order to keep things in balance.

B-3+3=2+3
(which is the same as)
B-0=5
(which is the same as)
B=5

Wasn't that easy? Algebra, or anything else, is only as hard as you "think" it is. We're not going to let people tell us that it's hard, because we've already seen that it's really pretty easy! We added 3 to both sides, because like we said earlier, the opposite of subtraction is addition. When you subtract from something, to undo what you did, you simply add it back! Think about that for a moment, then consider this: In the first problem we were adding 2 to F. In order to get rid of the 2 that we were adding we had to subtract it back out. Let's check the answer for our last problem by plugging 5 back in for B in the original problem.

5-3=2
(which is the same as)
2=2

Yes! We have a perfectly balanced result, which means that we have the right answer for B! Let's stop for a minute and consider 3 things before moving on.

1. We always want to keep our problems balanced on both sides of the equals sign!

2. To get rid of a number that we are adding or subtracting we have to do the opposite in order to remove it! If we are adding it to the letter, we subtract it back out. If we are subtracting it from the letter, we add it back in. We must do the same thing to both sides of the equals sign in order to keep the problem balanced!

3. We always want to get the letter on one side of the equals sign all by itself!


Now let's consider multiplication and division algebra problems. The opposite of multiplying something is to divide it. The opposite of dividing something is to multiply it.

When you multiply a division problem by the same number that you are using to divide with, the numbers will "cancel" each other out, or in other words, they undo each other. When you divide a multiplication problem by the same number that you are using to multiply with, the numbers will also "cancel" each other out. You use one to "undo" the other. Think for a moment about fractions. You can think of the number 3 as a fraction like this: 3/1. If you divide 3 by 3 like this: 3/3, the answer is 1. The same is true for any other number: 100/100=1, 45/45=1, and etc. Now think about 1 divided by 3 as a fraction like this: 1/3. If you multiply 1/3 by 3 the answer is 1. This is also true for any other number: 1/100 multiplied by 100 equals 1, 1/45 multiplied by 45 equals 1, and etc. Anytime you multiply or divide a number like this the answer will always be 1.

Let's talk about a multiplication algebra problem first. If you had 5xB and wanted to remove the 5, you would need to divide both sides of the equals sign by 5. Think about it. If B=2 you would have 5x2=10. If you divide 10 by 5 you get 2. If you divide (5x2) by 5 you also get 2. Both sides of the equals sign are balanced. Let's work a simple problem to see if this is really true.

2xC=6

Since this is a multiplication algebra problem, we'll use the opposite of multiplication, which is division, to "cancel" and get C on one side of the equals sign by itself.


(we cancel the 2's to get)
1xC=3
(which is the same as)
C=3

The way we got the 1 is because we know that 2/2=1. We also know that 1 multiplied by any other number is always that number. For example: 1x2=2 and 1x8=8. So it's not hard to see that 1xC is also going to equal C. Our final answer for C is C=3. Let's plug that back in for C in the original problem and see if we got the right answer.

2x3=6
(which is the same as)
6=6

Yes! A perfectly balanced problem, which means our solution for C was exactly right!

Now let's talk about a division algebra problem. If you had B/5 and wanted to remove the 5, you would need to multiply both sides of the equals sign by 5. Think about it. If B=10 you would have 10/5=2. If you multiply 2 by 5 you get 10. If you multiply (10/5) by 5 you also get 10. Both sides of the equals sign are balanced. Let's try a simple problem to see if this is really true.

D/3=2

We want to get D on one side of the equal sign by itself. Since this is a division algebra problem, we'll use the opposite of division, which is multiplication, to "cancel" and get rid of the 3.


(we cancel the 3's to get)
D/1=6
(which is the same as)
D=6

The way we got the 1 is because we know that 3x(1/3)=1. We also know that any number divided by 1 will always be that number. For example: 2/1=2 and 8/1=8. So it's not hard to see that D/1 is going to equal D. Another thing that we know is that anytime you multiply something by what you are dividing it by they will always "cancel" each other out or "undo" each other. Let's plug 6 back in for D in our original problem to see if this worked!

6/3=2
(which is the same as)
2=2

Again we have a perfectly balanced problem, which means we got the right answer for D!

If you don't understand how we canceled, please go back over what we talked about. Try a few problems on your own. Keep in mind that any division problem is the same as a fraction. 1/6 multiplied by 6 equals 1. 1/20 multiplied by 20 equals 1.

I'm going to throw a longer algebra problem at you now. I want to show you that longer algebra problems are just as easy as the shorter ones! Here's the problem.

2xE+3=11

This really isn't as bad as it looks! Let's do it one step at a time using everything we've learned so far. First let's get rid of the 3. Do you remember how to get rid of a number when you're adding it? Right! You subtract it!

2xE+3-3=11-3
(which is the same as)
2xE+0=8
(which is the same as)
2xE=8

See how simple that was? We've already gotten rid of one number. Now all we need to do to get E on one side of the equals sign by itself is to get rid of that 2! To get rid of a number when you're multiplying it, you have to divide by that same number just like we did before. Let's try it.


(we cancel the 2's to get)
1xE=4
(which is the same as)
E=4

There's our solution for E! Easy as pie! We knew that 2/2=1. We also knew that 1 multiplied by any number is going to be that same number. Now let's plug our answer for E back into the original problem to check and see if this is right!

2x4+3=11
(which is the same as)
8+3=11
(which is the same as)
11=11

Cool! We've just worked a long algebra problem and gotten the right answer! Now you're starting to see just how simple algebra can be! I probably should mention that most algebra problems don't use a multiplication sign. If you see a number beside a letter, it simply means to multiply the letter by that number. Here is an example.

2xG
(is the same as)
2G

5xT
(is the same as)
5T

The only difference is that you're getting rid of the multiplication sign. You could think of it like the sign is still there, but you just don't see it. I know it may sound silly, but you do it so that you won't mistake the multiplication sign for the letter "x", which is often used to represent unknown numbers in algebra problems.

Something else you should know about algebra is that if you see parenthesis, you MUST work that part of the problem first. The next thing you do is work any multiplication or division. The last thing you do is work any addition or subtraction. You work from left to right when you are doing each of these steps. Remember that (2+3)x2=10 is NOT the same as 2+(3x2)=8! Do you see how the parenthesis can make a BIG difference?

When working addition problems you can rearrange the numbers if it makes things easier. The answer will still be the same. For example: 2+4+8=14 is the same as 8+4+2=14. This is NOT true for subtraction! For example: 2-4-8=-10 is NOT the same as 8-4-2=2. You can also rearrange the numbers when working multiplication problems. The answer will still be the same. For example: 2x4x8=64 is the same as 8x4x2=64. This is NOT true for division! For example: 2/4/8=.0625 is NOT the same as 8/4/2=1. Do NOT mix addition and multiplication if you are going to rearrange! Remember that (2+3)x2=10 was NOT the same as 2+(3x2)=8!

Let's review everything one more time, then I'll give you a practice quiz to see what you've learned.

1. Algebra replaces unknown numbers in a problem with letters. You can use any letter to represent an unknown number. It doesn't matter.

2. Problems that have the correct answer are always perfectly balanced. Both sides of the equals sign are always equal to each other.

3. When you do something to one side of the equals sign, you must do the same exact thing to the other side in order to keep the problem balanced.

4. We always want to get the letter on one side of the equals sign all by itself.

5. We plug the answer we get for a letter back into the original problem to check and see if we got the right answer.

6. To get rid of a number, we have to do the opposite in order to remove it, remembering rule 3.

7. If we are adding a number to a letter, we subtract it back out to remove it, remembering rule 3.

8. If we are subtracting a number from a letter, we add it back in to remove it, remembering rule 3.

9. If we are multiplying the letter by a number, we have to divide it by the same number to get rid of it, remembering rule 3.

10. If we are dividing the letter by a number, we have to multiply it by the same number to get rid of it, remembering rule 3.

11. Longer algebra problems are just as easy as the shorter ones. We just have to work the problem one step at a time.

12. 6U is the same as 6xU, because most algebra problems don't use the multiplication sign.

13. Work what's inside of parenthesis first, multiplication and division second, and addition and subtraction last, working left to right each step.

14. You can rearrange addition or multiplication problems, but NEVER rearrange them if they are both together.


Make sure that you're comfortable with everything we've discussed. Reread the 14 rules above, and make sure you understand them. When you're ready, take the quiz below. You might want to have a piece of paper handy so that you can work out the problems before answering them.


QUIZ

Question 1:

L + 1 = 2

Answer:
A:  L = 3
B:  L = 4
C:  L = 1
D:  L = 2

Question 2:

M + 3 = 5

Answer:
A:  M = 2
B:  M = 8
C:  M = 1
D:  None of the above

Question 3:

4 + Q = 6

Answer:
A:  Q = 10
B:  Q = 3
C:  Q = 2
D:  Q = 6

Question 4:

J - 2 = 2

Answer:
A:  J = 0
B:  J = 4
C:  J = 2
D:  J = 1

Question 5:

K - 1 = 6

Answer:
A:  K = 6
B:  K = 5
C:  K = 4
D:  K = 7

Question 6:

Z - 5 = 5

Answer:
A:  Z = 0
B:  Z = 5
C:  Z = 1
D:  Z = 10

Question 7:

2 x T = 6

Answer:
A:  T = 2
B:  T = 3
C:  T = 4
D:  T = 8

Question 8:

W x 4 = 8

Answer:
A:  W = 2
B:  W = 4
C:  W = 12
D:  W = 1

Question 9:

H / 2 = 2

Answer:
A:  H = 1
B:  H = 2
C:  H = 4
D:  None of the above

Question 10:

3 x P + 4 = 10

Answer:
A:  P = 2
B:  P = 3
C:  P = 1
D:  P = 6




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