Hi. I am Larry. This is the video from Lesson 106B on my web site, Introduction to Algebra Part III. Be sure that you’ve seen the previous two lessons on my web site involving Algebra otherwise this lesson really wont make much sense to you.
For this lesson, I'm going to work on the problem and a little bit different on what we’ve looked at up until now. It can be a little intimidating.
Let’s just take a look on what we have here. We have 2/3 times x equals 14. Recall that since have no symbol between the 2/3 and the x. it means that we’ll multiply.
Now, right with that, many is doing just gets very uncomfortable with this. It’s important to understand that 2/3 is just a number. It happens to be a fraction but it’s a number. It’s no different that these have been 5 x.
There are two ways of solving this. Both of them will get the correct answer but one is a lot easier than the other. What we’ve learned up until now is that, if x is being multiplied by something, we will do the opposite which we know is divide so let’s take a look on how that works.
Since x is being multiplied with the 2/3, let’s go and do in reds. Since x is being multiplied by 2/3, I'm going to do the opposite which is to divide by 2/3. Now, since I did it on the left, I'm going to do the same thing on the right. Now we’ll already notice how this is getting a little messy. In a moment I’m going to show you a much easier way to do this but this does work and is most important to understand it.
On the left, 2/3 divided by 2/3 is just equals 1. That means we can say that they cancel this sense. 2/3 divided by 2/3 is 1 and that just leaves us with 1 x but we know it’s just x. Now look at what we have on the right, as messy as it looks, we have 14 divided by 2/3. Let’s write that in a more standard and clean way. I'm going to write 14 divided by 2/3.
Now, let’s keep going with that. We know that it’s a good idea to rewrite 14 as a fraction since we now in the world of fractions and we remember that the way to do that is by putting at over 1. We are always allowed do that, 14 divided by 2/3. I hopefully, you remember how to handle this. To divide by fraction, you multiply by the reciprocal of the fraction. So now we have x = 14 over 1 times, remember our middle dot is times, 3/2.
Now, we multiply across but before we do that, hopefully you remember that we’re allowed to using what we call cross cancelling. Basically, 2 goes into 2 once, 2 goes into 14 seven times. We now have 7 times 3 multiplied across which is 21 and in the denominator we just have 1 times 1 which is one and we don’t have to bother to write that.
So what we got is x = 21. Let’s just write it clean. Okay let’s check. I'll rewrite the original equation, and here is my checked value 2/3 times 21. Does that equal to14?
Now, since we are going to be multiplying a whole number times a fraction, remember it’s a good idea to rewrite that. Rewrite the 21 as a fraction. 2/3 times 21 over 1, does that equal 14? You could do some cross cancel, three it goes into three once and three goes into 21 seven times and multiply that across, we have 2 times 7 is 14 over 1 which is just 14. So that shows that 21 is the correct answer.
Now we did follow proper algebraic procedure and we did everything right and we got the right answer. Let’s go through a little bit of a shortcut in a sense of just an easier way to solving a problem that looks like this.
What I'm going to do, I’m going to rewrite. Let’s erase these, let’s see, I’m going to rewrite the original equation. And we’ll get an equivalent method, which is much faster and much easier. So my original equation was 2/3 x = 14.
By the way, it’s worth to knowing that, this x, very often we’ll kind of be written in the middle just like I did here. You just want to keep in mind that whenever you have something like that, the x is really in the numerator. Just think of it as x over 1. It’s 2/3 times some quantity, just keep that in mind.
Okay, here’s what we are going to do which is much easier on what we did before First of all I’m going to rewrite the original equation. You know if we have to but that make sense of second. I'm going to rewrite because I'm going to do a little trick or sorts.
What I have to see on myself at this point is how I get rid of these 2/3. I want x to be by itself but it is being multiplied by 2/3. Well, one of the tricks is that, if I'm multiplying by the reciprocal over here, make sure you see how the three’s will cancel each other out and the two’s will cancel each other out and remember it’s so much that they are cancelling but 3 dived 3 is 1. 2 divided 2 is one and where left with – We’re left with 1 x on the left which is just x again.
Since I've multiplied the left by 3/2, don’t get intimidated but that’s the number unlike any other number. I'm going to multiply the right by 3 x.
So were really done with x = 14 times 3/2. Recall that. We’re going put the 14 over 1 since we are on the world of fractions which is in cross canceling, 2 goes into 2 once, 2 goes into 14 seven times and just like before, we have x = 7 times 3 which is 21 over 1 or just x equals 21. I'll skip the check because it is exactly the same thing that we have the last time.
So the point to take away is that when we tried in the previous method. When we tried to divide both sides by 2/3 that did work, but it just got a little messy because down here, we have to divide by a fraction which ultimately meant and we have to change it to multiplication and multiply it by its reciprocal anyway.
So you see what we did we kind of skipped some steps. We skip that headache. Instead of having to work with division, what I did is like multiplying on both sides by the reciprocal. That allowing me to totally get rid of the 2/3 and then on the right we just had a simple multiplication problem. So when we see something like this, a fraction times a variable. Just multiplying both sides by the reciprocal and you will get your correct answer. Be sure to see the upcoming lessons where we’ll talk a lot of more about Algebra and we’ll start to get into working with signed numbers.
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