00:06Hi I’m Rob. Welcome to Math Antics.
00:09In our last video, we learned how to solve basic algebraic equations
00:13that only had one addition or one subtraction operation.
00:16In this video, we’ll focus on equations that have only one multiplication or one division operation.
00:22Now before we see some examples, do you remember the key strategy
00:26for solving an equation with an unknown value in it?
00:29Yep - we have to use arithmetic to rearrange the equation so that
00:34the unknown is all by itself on one side of the equal sign.
00:38And the most important thing to keep in mind while rearranging equations
00:42is that whenever we do something to one side of an equation,
00:45we have to do the same thing to the other side,
00:48or else, the other side might get jealous!
00:51“Hey, how come he got a cookie and I didn’t?”
00:57Actually, it’s to keep the equation in balance.
01:01Now remember from the last video,
01:02in equations where a number was being added to an unknown, we had to subtract that number from both sides.
01:09But when a number was being subtracted from the unknown, we had to add that number to both sides.
01:15And that makes sense because (as we learned in the video called “What is Arithmetic?”)
01:20addition and subtractions are Inverse Operations. They undo each other.
01:25Well guess what… Multiplication and Division are also inverse operations,
01:30so we can use them to undo each other too.
01:33If an unknown is being multiplied by a number, to undo that, we need to divide both sides by that number,
01:40but if an unknown is being divided by a number, to undo that, we need to multiply both sides by that number.
01:47Now don’t worry if that sounds a little confusing right now.
01:50It will make more sense after you’ve seen a few examples.
01:53Let’s start with this one: 3x = 15
01:57Ah, excuse me… I think you forget something.
02:00Didn’t you say that these equations were gonna have multiplication or division in them?
02:04But I don’t see ANY arithmetic operator at all in this equation.
02:08Actually, I think you forgot something that we learned in the video “What is Algebra?”
02:13You did watch that right?
02:15Uh…Oh… sure…sure, of course.…
02:17but I… ya know I… I just remembered…
02:20I have something I gotta do,
02:21I’ll… I’ll be right back…
02:23Well, I’m sure YOU remember that multiplication is the default operation in Algebra,
02:28so when you see a number and a symbol right next to each other like this, with no operation between them,
02:34it means they are being multiplied.
02:36So ‘3x’ is the same as 3 times x.
02:40Oh, and just a side note…
02:42since in multiplication, the order of the numbers doesn’t matter,
02:45you could switch the order and write ‘x3’,
02:49but it’s customary to always list the known number first and the unknown number second.
02:55Alright, but we need to solve this equation, right?
02:58That means we need to get the unknown ‘x’ all by itself on one side of the equal sign.
03:03Right now, the ‘x’ is not by itself because it’s being multiplied by 3.
03:08So, to undo that operation, we need to divide that side by 3.
03:13In Algebra, we almost always write division in fraction form,
03:17so to divide this side by 3, we just write a fraction line under it, and we put a 3 below the line.
03:24There, this means 3 times x divided by 3.
03:29Ah! - But don’t forget our rule for rearranging equations.
03:33We have to do the exact same thing to the other side to keep the equation balanced.
03:38That’s better. Now both sides are being divided by 3.
03:42The next step is to simplify.
03:43The 3 on the top and the 3 on the bottom of this side cancel, because 3 divided by 3 would just be 1.
03:50This is just like canceling common factors when you are simplifying a fraction.
03:55That leaves us with just ‘x’ on this side.
03:58And on the other side, we have 15 divided by 3, which simplifies to 5.
04:03There… we’ve solved our equation by changing it into the simplified form: x = 5.
04:09Let’s try another one like that: 12x = 96.
04:14In this problem, the unknown is being multiplied by 12, so to get the ‘x’ all by itself,
04:19we’re going to need to divide both sides of the equation by 12.
04:23On the first side, the 12 on top and the 12 on bottom cancel out, leaving just ‘x’ on that side.
04:29And on the other side, we need to divide 96 by 12.
04:33You might be able to do that by memory,
04:35but if not, you can use a calculator to divide.
04:3896 divided by 12 is 8. So in this problem, x = 8.
04:44That’s pretty easy, isn’t it?
04:46Are your ready to try a division problem now?
04:48Here we have x ÷ 2 = 3.
04:52Now when you see division written like this (from left to right with the traditional division symbol)
04:57I want you to re-write it using the fraction form for division.
05:01And that’s because it’s much easier to cancel common factors
05:05and simplify your equation when you use the fraction form.
05:09Now that we have it re-written, let’s solve it.
05:12We can see that the unknown is not by itself because it is being divided by 2.
05:17How can get get rid of (or undo) that division?
05:20Yep… we can undo division with multiplication.
05:24So we need to multiply BOTH sides of the equation by 2.
05:28Instead of writing the multiplication sign,
05:30I’m using the parentheses notation that we learned about in the video called “What Is Algebra?”
05:35Remember, the multiplication is just implied.
05:39On the first side, the 2 on top cancels out the 2 on the bottom,
05:44since 2 divided by 2 is just 1.
05:47And I know what some of you are thinking…
05:49“How is there a 2 on top? The 2 looks like it’s really in the middle …kind of like how a mixed number looks.”
05:55That’s true, but don’t confuse this with a mixed number!
05:59Mixed numbers involve addition,
06:01but the parentheses let you know that the 2 and the (x over 2) are being multiplied,
06:06since multiplication is the default operation.
06:09Okay, so it’s not a mixed number, but how is the 2 on top?
06:13Well, do you remember how you can turn any number into a fraction
06:17just by making 1 the bottom number?
06:19That means that 2 is the same as 2 over 1.
06:23Ah… now you can see that the 2 really is on top.
06:26It’s just that we don’t usually show the 1 on the bottom.
06:30Alright then… so the ‘2’s cancel, leaving the ‘x’ all by itself on this side.
06:35And on the other side, we have 3 times 2, which is just 6.
06:39So in this problem, x = 6.
06:41That’s not too hard either!
06:43Let’s try another one: x over 10 = 15.
06:47In this problem, since the x is being divided by 10, to get it by itself,
06:52we’re going to need to multiply both sides of the equation by 10.
06:57On the first side, the ’10’s cancel, leaving ‘x’ all by itself.
07:01And on the other side, we have 15 times 10, which is 150.
07:06So our answer is x = 150.
07:10Great! That’s how you solve simple equation where an unknown is being multiplied by a number or divided by a number.
07:17But, just like with subtraction in the last video, with division,
07:22there’s a tricky variation that I need to tell you about.
07:25What if you have an equation where a number is being divided by an unknown?
07:31Since division does not have the commutative property,
07:34x over 4 is NOT the same thing as 4 over x.
07:39So what do we do if the unknown is on the bottom?
07:42…like in this problem: 4 over x = 2
07:46Well, you’re first thought might be to multiplying both sides by 4,
07:50but that won’t help us here, because both of the ‘4’s would be on top,
07:55so they wouldn’t cancel each other out.
07:57Instead, what we need to do is multiply both sides by ‘x’.
08:01Watch what happens then…
08:03The ‘x’s on this side of the equation will cancel.
08:07Yep! You can cancel unknowns and variables exactly like you can regular numbers.
08:13That will leave us with just 4 on this side of the equation,
08:16and on the other side, we have 2 times x or 2x.
08:22True… that didn’t solve the equation!
08:24But it did get rid of the tricky ‘x’ on the bottom
08:27and it changed our equation into a problem that we already know how to solve.
08:31Now, to get the ‘x’ all by itself, we just need to divide both sides of the equation by 2.
08:37On the first side, we have 4 divided by 2, which is 2,
08:40and on the other side, the 2 over 2 cancels and we are left with just x.
08:45So now we know that x = 2.
08:49Okay… so now that you’ve watched these first three Math Antics Algebra videos,
08:53you should be able to solve any simple one-step equation involving
08:57addition, subtraction, multiplication or division, right?
09:01Well… not unless you practice!
09:03To really learn how to solve equations,
09:05you have to try a lot of problems on your own to make sure that you really understand how to do it.
09:10And if you’re still confused, try re-watching these videos a few times since they cover so much information.
09:17As always, thanks for watching Math Antics, and I’ll see ya next time!
09:21Learn more at www.mathantics.com
09:24Hey, I watched that video you… mentioned…
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