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