Algebra seems to bother people because instead of numbers we sometimes use letters. What no one ever tells people is that they were doing algebra back in 2nd grade. Consider the following problem, found in 2nd grade math books all over the world.
__
Fill in the blank 2 + | | = 5
|__|
That problem doesn't bother anyone. Obviously the answer is 3. Consider the following problem.
Solve for x. 2 + x = 5. This is exactly the same problem. The only difference is that a letter (a variable) is now in place of the box. Variables are simply symbols (like boxes) that are used to take the place of unknown quantities. Because we may have equations using more than one unknown, it is easier to use letters rather than boxes and triangles.
Let's solve this problem using the actual method we subconciously use. We solve this by subtracting 2 from 5. Why? Well, the variable is what we want to solve for. But something is being done to it. It has a 2 being added to it. We want to undo that so it will be by itself. We can't just subtract 2 from the left side. We have two things that are equal, "2 + x" and "5." If they are equal, then they will still be equal if we subtract 2 from each one.
2 + x - 2 = 5 - 2Let's look at a similar problem.
Solve for x : 2x = 6 (Keep in mind that when no symbol is given between the coefficient [the number in front of the x] and the variable, it is understood to be multiplication.)
We again have something being done to our variable. This time it is multiplication by 2. To undo that, we will divide by 2. This gives
2x 6
------ = -----
2 2
Since 2 divided by 2 is 1 and 6 divided by 2 is 3 we have x = 3.
One problem that a lot of people have with algebra is similar to the one they have with word problems. The first word problems you deal with are so simple that very little understanding of the thought process involved in solving the problem is really necessary (If Johnny has two apples and is given another, how many does he have?). But as the problems get more complicated (one train leaves Los Angeles, etc.) the need to understand the thought process becomes greater but no experience in that understanding has been developed.
The same is true in algebra. In the 2nd grade problem above, the answer is so obvious that you don't really need to understand the solution process. But when the problems get harder, the process is necessary.
Let's try another problem, that is a little more complicated.
Solve for x: 2x + 5 = 11. As in the second grade problem, we want to know the value that goes in the blank, except this time the "blank" is an x, to make the statement true. Notice that the left hand side of the equation has two terms (terms are quantities that are being added together). We have a "2x" and a "5." One of those terms has the variable in it. We want to get that term isolated on the left hand side. To do that, we reverse what is being done to it. It is having a 5 added to it. To reverse addition by 5 we subtract 5. Of course, doing that to only one side of an equation would be invalid, because the two sides are equal. If we take 5 away from one side only, they will no longer be equal, hence we take 5 away from each.
Thus we have 2x + 5 - 5 = 11 - 5 or 2x = 6.
Now, again, remember what we are trying to do. We want to isolate the variable. It still has something being done to it. It is being multiplied by 2. To reverse multiplication by 2, we divide by 2. Again, we do it to both sides for the same reason as before.
2x 6
This gives ---- = ---- . Since 2 divided by 2 is 1 and 6 divided
2 2
by 2 is 3, we get 1x = 3 but when the coefficient (the number in front of the variable) is a 1 we don't bother writing it. Thus we have our solution
x = 3.We can check to see that having x = 3 makes the original statement true by replacing x with 3 in 2x + 5 = 11. Remember, the x is not turning into a number instead of a letter. It has always been a number. Since 2 times 3 is 6 and 6 + 5 is 11 we found the correct answer.
We'll now make it a little harder.
Solve for x: 3x
------ - 6 = 8
5
We'll start exactly the same as with the last problem by noting that on the left hand side we have two terms. Again, terms are things that are being added (or subtracted) together. We want to get the term with the variable by itself so we undo what is being done to it. It is having 6 subtracted from it. So we add 6 to both sides.
This gives
3x
------ - 6 + 6 = 8 + 6
5
or
3x
------ = 14
5
We have a couple of ways of looking at the left hand side. We can handle it as a two step process or one. Let's do it as a two step process. First note that the variable is being divided by 5. To undo division by 5 we multiply by 5.
This gives 3x
5 ----- = 14(5) The parentheses are just to keep the
5 numbers from running into each other.
or 3x = 70. Now we have our variable multiplied by 3. To undo
that we divide by 3 and have
3x 70
------ = ---
3 3
or
70
x = ----
3
The other way the problem
3x
------ = 14
5
could be done is to look at it as though the x is being multiplied
by
3
---
5.
3
To undo that we divide by ---
5
Keep in mind that in a fraction, the denominator (bottom number) tells the number of pieces the whole is divided into. So it is a dividing number. So when we are reversing things, it will become a multiplying number. Likewise, the numerator (the top number) tells how many of something we have so it is a multiplying number. Thus, when reversing things, it will become a dividing number. Therefore,
dividing by 3 is the same as multiplying by 5 .
--- ---
5 3
Thus we have
5 3x 5
- ------ = 14 -
3 5 3
70
or x = ----
3
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