Hi. I'm Larry. This is the video for lesson 106A on my web site Introduction to Algebra Part II. Be sure that you’ve watched the previous lesson on Algebra lesson 10five because this lesson assumes that you have. If you haven’t watched that lesson, a lot of what I am doing, you may not understand then I may go too quickly.
In the last lesson, we solved an Algebra problem involving subtraction. For this lesson let’s do one involving addition. Lets solve x + 4 = 9.
Now recall from the previous lesson that we need to get x by itself. We look to see what’s being done to the x and then we are going to do the opposite to both sides.
Here, in can see that four is being added to x so we’ll do the opposite which is subtract form and remember that since I did that on the left, I have much to do the same thing on the right. Immediately do the same thing on the right without even thinking about it because we need to keep the equation in balance. Whatever we do to the left, we have to do to the right.
I only used the red just to show what I did, though it feels that you have to use color pencils or color pens or anything like that.
Now, on the right we have the nine minus four that’s just five. I'll bring them by equal sign. Now on the left recall how this works. The plus four and the minus four, we consider they cancel out but what really happens is that plus four and minus four equals zero and in addition, in the world of addition, we don’t bother to right the zero. So on the left, where left with just x and we have our answer, x equals five.
Get into the habit of checking like we spoke about it in the previous lesson of copying the original equation. Here is my test value and we think that x equals five. So does the left side equal the right side? Five plus four is a nine which it does equal nine. So that shows that the answer five is correct. It is the value of x that makes the original equation true.
Now, let’s take a look at a different problem. Now I’m going to do a problem involving multiplication. For this problem I have 3 x = 18. Now let’s just take a minute to talk about this. Remember the last lesson we said that x does not mean multiplying. x does not, or knowing using x to mean multiplication. We are using x just as a variable. It’s an unknown value that we need to solve for.
Now, we’ve best said, don’t get confused. Here, I have three x, and recall from previous lessons that if there’s no symbol in between two things, we mean multiply. We could also mean multiplying if I cut a little middle dot in between the three and the x so don’t get confused on the thing just to because an x is use for general multiplication. If this have been three a equals 18, remember we can use any letter if I had that three a. We would also be implying multiplication because there is no symbol between the three and the a. so just make sure you’re clear about that.
When we had the x here, it had nothing to do with multiplication. It’s just an unknown. This is also is an unknown but I know that we are dealing with multiplication because there’s no symbol in between the three and the x.
So we are dealing with three times some unknown equals 18. Now, this is a little bit different than what you’ve seen. So remember our flat process. We are starting with x and we look to see what’s being done to it.
Here, what’s being done to it is, it’s being multiplied by three. Now how do undo multiplying by three? We know that the opposite of multiplying is divide. So to undo multiplying by three, we are going to divide it by three. So I'm going to divide by three here and we will always without even thinking, do the same thing on the right. Whatever you do to the left, you must always do to the right to keep the equation in balance. Don’t even think about it, the more we could do something on the left, immediately do the same thing on the right.
On the right, we have 18 divided three, make sure you remember that it’s always top number divided by bottom number. 18 divided three is six. I brought down my equal sign.
Now on the left, what most students will do is the perfect cross out of the three and the three say they cancel. That is true but it’s very important to understand what’s really happening here. We are dealing with the three divided three. Now, think about that for a minute. three divided three is one. So it’s not just minus it’s not that reasonably disappear. It’s that we are dealing with one. three divided three is one. So what we really have on the left is one x equals 6.
Now, we know that in the world of multiplication, our number that doesn’t matter is one. We call that the identity element but it doesn’t matter what you call it. We know that one is a number that we don’t really have to write because anything times one is itself.
So it’s not just that the three is magically vanished. Instead, three divided three equals 1 and we don’t have to bother right. So it is typically just right x. So we got x equal to six. We believe that that is the value of x which will make the original equation true? Let’s check.
3 x = 18. I rewrote my original equation. Now I’m going to substitute. Now, here when I substitute my checked value I have to put it in parenthesis or I have to put on the middle a little dot there it show multiplication. If I don’t, it’s going to look up the number 36 and that’s not what we mean. We mean the three times six. Does that equal 18? I we know that three times six is 18. 18 is equal to18 so that shows that our value of 6 is correct in the original equation.
Now, let’s take a look at another example. In this example, I have the division problem. Now, this maybe a little bit overwhelming but don’t get confused. Its x divided by seven = five. So some number divided by seven and we need for that to equal five. We need to figure out what value of x will make this true. Again, we look to see what’s being done to the x and we’ll do the opposite. Here, x is being divided by seven so instead what we’re going to do to the opposite is divided by seven which we know is multiplying by seven.
Now, very often it will be written like this, kind of like with big seven like that. Hold that thought for one second, I did it on the left so remember that I must immediately do it on the right. The right side is easy, seven times five is 35.
Now, one way to think about what’s going on, on the left is just simply that the seven is canceled. Both of these things will just cross out the seven’s and just write x as their answer. After all seven done, that is how it works. But it’s important to understand why I will have to cancel this seven’s.
Be sure to remember that when I have a whole number, I'm always allowed to write it over one. So make sure that you see that this seven could be rewritten as seven over one and then times x over seven. Everything I’ve shown you now is unnecessary but I'm trying to show you why we are allowed to just cross out this sevens.
We know that when we multiply, we multiply across on top we have seven x; on the right have the seven times one which is seven. And now, we can see like we’ve seen before, the seven is just cancelled because seven over seven equals one and x = 35.
Okay so all of these was unnecessary. We could have just crossed out this seven instead of right of the back. That is alternately what we’ll do but make sure you see that it’s not just magic. It’s because this seven over here, even if it’s written really big next to this fraction. Even if it’s written big, we know it’s really in the numerator and that gives us the right to cross out the sevens.
Okay to check, I’ll rewrite the original equation. I’ll substitute what I think is the correct answer x = 35. I’ll put in parenthesis just to show that’s my check value. Does 35 divided seven equal five? Yes it does 35 divided by seven is five which equals five. So that shows that our answer of 35 is correct.
Okay so this is just part two of our introduction to Algebra. Be sure to look at the upcoming lessons and if you take this further.
Transcription by:
Scribe4you Transcription Services