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2nd SESSION part 2: How can I solve an equation?

 Now, you can solve the equations proposed. Be brave!!

TRANSCRIPTION

 0:06
Hi I’m Rob. Welcome to Math Antics.
0:09
In our last video, we learned how to solve basic algebraic equations
0:13
that only had one addition or one subtraction operation.
0:16
In this video, we’ll focus on equations that have only one multiplication or one division operation.
0:22
Now before we see some examples, do you remember the key strategy
0:26
for solving an equation with an unknown value in it?
0:29
Yep - we have to use arithmetic to rearrange the equation so that
0:34
the unknown is all by itself on one side of the equal sign.
0:38
And the most important thing to keep in mind while rearranging equations
0:42
is that whenever we do something to one side of an equation,
0:45
we have to do the same thing to the other side,
0:48
or else, the other side might get jealous!
0:51
“Hey, how come he got a cookie and I didn’t?”
0:57
Actually, it’s to keep the equation in balance.
1:01
Now remember from the last video,
1:02
in equations where a number was being added to an unknown, we had to subtract that number from both sides.
1:09
But when a number was being subtracted from the unknown, we had to add that number to both sides.
1:15
And that makes sense because (as we learned in the video called “What is Arithmetic?”)
1:20
addition and subtractions are Inverse Operations. They undo each other.
1:25
Well guess what… Multiplication and Division are also inverse operations,
1:30
so we can use them to undo each other too.
1:33
If an unknown is being multiplied by a number, to undo that, we need to divide both sides by that number,
1:40
but if an unknown is being divided by a number, to undo that, we need to multiply both sides by that number.
1:47
Now don’t worry if that sounds a little confusing right now.
1:50
It will make more sense after you’ve seen a few examples.
1:53
Let’s start with this one: 3x = 15
1:57
Ah, excuse me… I think you forget something.
2:00
Didn’t you say that these equations were gonna have multiplication or division in them?
2:04
But I don’t see ANY arithmetic operator at all in this equation.
2:08
Actually, I think you forgot something that we learned in the video “What is Algebra?”
2:13
You did watch that right?
2:15
Uh…Oh… sure…sure, of course.…
2:17
but I… ya know I… I just remembered…
2:20
I have something I gotta do,
2:21
I’ll… I’ll be right back…
2:23
Well, I’m sure YOU remember that multiplication is the default operation in Algebra,
2:28
so when you see a number and a symbol right next to each other like this, with no operation between them,
2:34
it means they are being multiplied.
2:36
So ‘3x’ is the same as 3 times x.
2:40
Oh, and just a side note…
2:42
since in multiplication, the order of the numbers doesn’t matter,
2:45
you could switch the order and write ‘x3’,
2:49
but it’s customary to always list the known number first and the unknown number second.
2:55
Alright, but we need to solve this equation, right?
2:58
That means we need to get the unknown ‘x’ all by itself on one side of the equal sign.
3:03
Right now, the ‘x’ is not by itself because it’s being multiplied by 3.
3:08
So, to undo that operation, we need to divide that side by 3.
3:13
In Algebra, we almost always write division in fraction form,
3:17
so to divide this side by 3, we just write a fraction line under it, and we put a 3 below the line.
3:24
There, this means 3 times x divided by 3.
3:29
Ah! - But don’t forget our rule for rearranging equations.
3:33
We have to do the exact same thing to the other side to keep the equation balanced.
3:38
That’s better. Now both sides are being divided by 3.
3:42
The next step is to simplify.
3:43
The 3 on the top and the 3 on the bottom of this side cancel, because 3 divided by 3 would just be 1.
3:50
This is just like canceling common factors when you are simplifying a fraction.
3:55
That leaves us with just ‘x’ on this side.
3:58
And on the other side, we have 15 divided by 3, which simplifies to 5.
4:03
There… we’ve solved our equation by changing it into the simplified form: x = 5.
4:09
Let’s try another one like that: 12x = 96.
4:14
In this problem, the unknown is being multiplied by 12, so to get the ‘x’ all by itself,
4:19
we’re going to need to divide both sides of the equation by 12.
4:23
On the first side, the 12 on top and the 12 on bottom cancel out, leaving just ‘x’ on that side.
4:29
And on the other side, we need to divide 96 by 12.
4:33
You might be able to do that by memory,
4:35
but if not, you can use a calculator to divide.
4:38
96 divided by 12 is 8. So in this problem, x = 8.
4:44
That’s pretty easy, isn’t it?
4:46
Are your ready to try a division problem now?
4:48
Here we have x ÷ 2 = 3.
4:52
Now when you see division written like this (from left to right with the traditional division symbol)
4:57
I want you to re-write it using the fraction form for division.
5:01
And that’s because it’s much easier to cancel common factors
5:05
and simplify your equation when you use the fraction form.
5:09
Now that we have it re-written, let’s solve it.
5:12
We can see that the unknown is not by itself because it is being divided by 2.
5:17
How can get get rid of (or undo) that division?
5:20
Yep… we can undo division with multiplication.
5:24
So we need to multiply BOTH sides of the equation by 2.
5:28
Instead of writing the multiplication sign,
5:30
I’m using the parentheses notation that we learned about in the video called “What Is Algebra?”
5:35
Remember, the multiplication is just implied.
5:38
Now to simplify…
5:39
On the first side, the 2 on top cancels out the 2 on the bottom,
5:44
since 2 divided by 2 is just 1.
5:47
And I know what some of you are thinking…
5:49
“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.”
5:55
That’s true, but don’t confuse this with a mixed number!
5:59
Mixed numbers involve addition,
6:01
but the parentheses let you know that the 2 and the (x over 2) are being multiplied,
6:06
since multiplication is the default operation.
6:09
Okay, so it’s not a mixed number, but how is the 2 on top?
6:13
Well, do you remember how you can turn any number into a fraction
6:17
just by making 1 the bottom number?
6:19
That means that 2 is the same as 2 over 1.
6:23
Ah… now you can see that the 2 really is on top.
6:26
It’s just that we don’t usually show the 1 on the bottom.
6:30
Alright then… so the ‘2’s cancel, leaving the ‘x’ all by itself on this side.
6:35
And on the other side, we have 3 times 2, which is just 6.
6:39
So in this problem, x = 6.
6:41
That’s not too hard either!
6:43
Let’s try another one: x over 10 = 15.
6:47
In this problem, since the x is being divided by 10, to get it by itself,
6:52
we’re going to need to multiply both sides of the equation by 10.
6:57
On the first side, the ’10’s cancel, leaving ‘x’ all by itself.
7:01
And on the other side, we have 15 times 10, which is 150.
7:06
So our answer is x = 150.
7:10
Great! That’s how you solve simple equation where an unknown is being multiplied by a number or divided by a number.
7:17
But, just like with subtraction in the last video, with division,
7:22
there’s a tricky variation that I need to tell you about.
7:25
What if you have an equation where a number is being divided by an unknown?
7:31
Since division does not have the commutative property,
7:34
x over 4 is NOT the same thing as 4 over x.
7:39
So what do we do if the unknown is on the bottom?
7:42
…like in this problem: 4 over x = 2
7:46
Well, you’re first thought might be to multiplying both sides by 4,
7:50
but that won’t help us here, because both of the ‘4’s would be on top,
7:55
so they wouldn’t cancel each other out.
7:57
Instead, what we need to do is multiply both sides by ‘x’.
8:01
Watch what happens then…
8:03
The ‘x’s on this side of the equation will cancel.
8:07
Yep! You can cancel unknowns and variables exactly like you can regular numbers.
8:13
That will leave us with just 4 on this side of the equation,
8:16
and on the other side, we have 2 times x or 2x.
8:22
True… that didn’t solve the equation!
8:24
But it did get rid of the tricky ‘x’ on the bottom
8:27
and it changed our equation into a problem that we already know how to solve.
8:31
Now, to get the ‘x’ all by itself, we just need to divide both sides of the equation by 2.
8:37
On the first side, we have 4 divided by 2, which is 2,
8:40
and on the other side, the 2 over 2 cancels and we are left with just x.
8:45
So now we know that x = 2.
8:49
Okay… so now that you’ve watched these first three Math Antics Algebra videos,
8:53
you should be able to solve any simple one-step equation involving
8:57
addition, subtraction, multiplication or division, right?
9:01
Well… not unless you practice!
9:03
To really learn how to solve equations,
9:05
you have to try a lot of problems on your own to make sure that you really understand how to do it.
9:10
And if you’re still confused, try re-watching these videos a few times since they cover so much information.
9:17
As always, thanks for watching Math Antics, and I’ll see ya next time!
9:21
Learn more at www.mathantics.com
9:24
Hey, I watched that video you… mentioned…
9:27
Hello?… Hello?
9:30
Where am I?

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