Hi I’m Rob. Welcome to Math Antics.

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In our last video, we learned how to solve basic algebraic equations

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that only had one addition or one subtraction operation.

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In this video, we’ll focus on equations that have only one multiplication or one division operation.

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Now before we see some examples, do you remember the key strategy

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for solving an equation with an unknown value in it?

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Yep - we have to use arithmetic to rearrange the equation so that

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the unknown is all by itself on one side of the equal sign.

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And the most important thing to keep in mind while rearranging equations

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is that whenever we do something to one side of an equation,

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we have to do the same thing to the other side,

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or else, the other side might get jealous!

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“Hey, how come he got a cookie and I didn’t?”

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Actually, it’s to keep the equation in balance.

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Now remember from the last video,

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in equations where a number was being added to an unknown, we had to subtract that number from both sides.

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But when a number was being subtracted from the unknown, we had to add that number to both sides.

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And that makes sense because (as we learned in the video called “What is Arithmetic?”)

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addition and subtractions are Inverse Operations. They undo each other.

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Well guess what… Multiplication and Division are also inverse operations,

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so we can use them to undo each other too.

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If an unknown is being multiplied by a number, to undo that, we need to divide both sides by that number,

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but if an unknown is being divided by a number, to undo that, we need to multiply both sides by that number.

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Now don’t worry if that sounds a little confusing right now.

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It will make more sense after you’ve seen a few examples.

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Let’s start with this one: 3x = 15

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Ah, excuse me… I think you forget something.

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Didn’t you say that these equations were gonna have multiplication or division in them?

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But I don’t see ANY arithmetic operator at all in this equation.

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Actually, I think you forgot something that we learned in the video “What is Algebra?”

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You did watch that right?

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Uh…Oh… sure…sure, of course.…

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but I… ya know I… I just remembered…

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I have something I gotta do,

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I’ll… I’ll be right back…

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Well, I’m sure YOU remember that multiplication is the default operation in Algebra,

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so when you see a number and a symbol right next to each other like this, with no operation between them,

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it means they are being multiplied.

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So ‘3x’ is the same as 3 times x.

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Oh, and just a side note…

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since in multiplication, the order of the numbers doesn’t matter,

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you could switch the order and write ‘x3’,

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but it’s customary to always list the known number first and the unknown number second.

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Alright, but we need to solve this equation, right?

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That means we need to get the unknown ‘x’ all by itself on one side of the equal sign.

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Right now, the ‘x’ is not by itself because it’s being multiplied by 3.

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So, to undo that operation, we need to divide that side by 3.

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In Algebra, we almost always write division in fraction form,

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so to divide this side by 3, we just write a fraction line under it, and we put a 3 below the line.

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There, this means 3 times x divided by 3.

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Ah! - But don’t forget our rule for rearranging equations.

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We have to do the exact same thing to the other side to keep the equation balanced.

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That’s better. Now both sides are being divided by 3.

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The next step is to simplify.

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The 3 on the top and the 3 on the bottom of this side cancel, because 3 divided by 3 would just be 1.

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This is just like canceling common factors when you are simplifying a fraction.

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That leaves us with just ‘x’ on this side.

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And on the other side, we have 15 divided by 3, which simplifies to 5.

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There… we’ve solved our equation by changing it into the simplified form: x = 5.

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Let’s try another one like that: 12x = 96.

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In this problem, the unknown is being multiplied by 12, so to get the ‘x’ all by itself,

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we’re going to need to divide both sides of the equation by 12.

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On the first side, the 12 on top and the 12 on bottom cancel out, leaving just ‘x’ on that side.

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And on the other side, we need to divide 96 by 12.

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You might be able to do that by memory,

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but if not, you can use a calculator to divide.

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96 divided by 12 is 8. So in this problem, x = 8.

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That’s pretty easy, isn’t it?

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Are your ready to try a division problem now?

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Here we have x ÷ 2 = 3.

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Now when you see division written like this (from left to right with the traditional division symbol)

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I want you to re-write it using the fraction form for division.

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And that’s because it’s much easier to cancel common factors

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and simplify your equation when you use the fraction form.

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Now that we have it re-written, let’s solve it.

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We can see that the unknown is not by itself because it is being divided by 2.

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How can get get rid of (or undo) that division?

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Yep… we can undo division with multiplication.

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So we need to multiply BOTH sides of the equation by 2.

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Instead of writing the multiplication sign,

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I’m using the parentheses notation that we learned about in the video called “What Is Algebra?”

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Remember, the multiplication is just implied.

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On the first side, the 2 on top cancels out the 2 on the bottom,

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since 2 divided by 2 is just 1.

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And I know what some of you are thinking…

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“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.”

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That’s true, but don’t confuse this with a mixed number!

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Mixed numbers involve addition,

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but the parentheses let you know that the 2 and the (x over 2) are being multiplied,

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since multiplication is the default operation.

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Okay, so it’s not a mixed number, but how is the 2 on top?

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Well, do you remember how you can turn any number into a fraction

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just by making 1 the bottom number?

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That means that 2 is the same as 2 over 1.

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Ah… now you can see that the 2 really is on top.

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It’s just that we don’t usually show the 1 on the bottom.

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Alright then… so the ‘2’s cancel, leaving the ‘x’ all by itself on this side.

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And on the other side, we have 3 times 2, which is just 6.

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So in this problem, x = 6.

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That’s not too hard either!

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Let’s try another one: x over 10 = 15.

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In this problem, since the x is being divided by 10, to get it by itself,

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we’re going to need to multiply both sides of the equation by 10.

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On the first side, the ’10’s cancel, leaving ‘x’ all by itself.

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And on the other side, we have 15 times 10, which is 150.

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So our answer is x = 150.

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Great! That’s how you solve simple equation where an unknown is being multiplied by a number or divided by a number.

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But, just like with subtraction in the last video, with division,

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there’s a tricky variation that I need to tell you about.

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What if you have an equation where a number is being divided by an unknown?

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Since division does not have the commutative property,

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x over 4 is NOT the same thing as 4 over x.

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So what do we do if the unknown is on the bottom?

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…like in this problem: 4 over x = 2

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Well, you’re first thought might be to multiplying both sides by 4,

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but that won’t help us here, because both of the ‘4’s would be on top,

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so they wouldn’t cancel each other out.

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Instead, what we need to do is multiply both sides by ‘x’.

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Watch what happens then…

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The ‘x’s on this side of the equation will cancel.

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Yep! You can cancel unknowns and variables exactly like you can regular numbers.

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That will leave us with just 4 on this side of the equation,

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and on the other side, we have 2 times x or 2x.

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True… that didn’t solve the equation!

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But it did get rid of the tricky ‘x’ on the bottom

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and it changed our equation into a problem that we already know how to solve.

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Now, to get the ‘x’ all by itself, we just need to divide both sides of the equation by 2.

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On the first side, we have 4 divided by 2, which is 2,

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and on the other side, the 2 over 2 cancels and we are left with just x.

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So now we know that x = 2.

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Okay… so now that you’ve watched these first three Math Antics Algebra videos,

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you should be able to solve any simple one-step equation involving

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addition, subtraction, multiplication or division, right?

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Well… not unless you practice!

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To really learn how to solve equations,

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you have to try a lot of problems on your own to make sure that you really understand how to do it.

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And if you’re still confused, try re-watching these videos a few times since they cover so much information.

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As always, thanks for watching Math Antics, and I’ll see ya next time!

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Learn more at www.mathantics.com

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Hey, I watched that video you… mentioned…