Hi! This is Rob. Welcome to Math Antics.
In this lesson, we're going to learn some really important things
about a whole branch of math called Algebra.
The first thing you need to know is that Algebra is a lot like arithmetic.
It follows all the rules of arithmetic
and it uses the same four main operations that arithmetic is built on:
addition, subtraction, multiplication and division.
But Algebra introduces a new element...
the element of the unknown.
When you were learning arithmetic,
the only thing that was ever unknown was the answer to the problem.
For example, you might have the problem 1 + 2 = what?
The answer isn't known until you go ahead and do the arithmetic.
Now the important thing about Algebra is that when we don't know what a number is yet,
we use a symbol in its place.
that symbol is usually just any letter of the alphabet.
A really popular letter to choose is the letter 'x'.
So in arithmetic, we would just leave the problem like this: 1 + 2 = "blank"
and we'd write in the answer when we did the addition.
But in Algebra, we'd write it like this: 1 + 2 = x
The 'x' is a place holder that stands for the number that we don't know yet.
What we have here is a very basic algebraic equation.
An equation is just a mathematical statement that two things are equal.
An equation says, "the things on this side of the equal sign
have the same value as the things on the other side of the equal sign."
In this case, our equation is telling us that the known values on this side (1+2)
are equal to what's on the other side,
which happens to be the unknown value that we are calling 'x'.
One of the main goals in Algebra
is to figure out what the unknown values in equations are.
And when you do that, it's called "solving the equations".
In this equation, it's pretty easy to see that the unknown value is just 3.
All you have to do is actually add the 1 and 2 together on this side of the equation
and it turns into 3 = x, which is the same as x = 3.
So now we know what 'x' is! It's just 3.
That almost seems too easy, doesn’t it?
And that's why in Algebra, you are usually given an equation in a more complicated form
like this: x - 2 = 1.
This is exactly the same equation as 1 + 2 = x,
but it has been rearranged so that it's not quite as easy to tell what 'x' is.
So in Algebra, solving equations is a lot like a game
where you are given mixed-up, complicated equations,
and it's your job to simplify them and rearrange them
until it is a nice simple equation (like x = 3)
where it's easy to tell what the unknown values are.
We're going to learn a lot more about
how you actually do that (how you solve equations)
in the next several videos,
but for now, let's learn some important rules about
how symbols can and can't be used in algebraic equations.
The first rule you need to know is that the same symbol (or letter)
can be used in different algebra problems to stand for different unknown values.
For example, in the problem we just solved,
the letter 'x' was used to stand for the number 3, right?
But 'x' could stand for a different number in a different problem.
Like, if someone asks us to solve the equation, 5 + x = 10.
In order for the two sides of this equation to be equal,
'x' must have the value '5' in this problem, because 5 + 5 = 10.
So 'x' (or any other symbol) can stand for different values in different problems.
but what's NOT okay is for a symbol
to stand for different values in the same problem at the same time!
For example, what if you had the equation: x + x = 10?
This equation says that if we add 'x' to 'x' we will get 10.
And there are a lot of different numbers we could add together to get 10, like 6 and 4.
But, if we had the first 'x' stand for 6 and the second 'x' stand for 4,
then 'x' would stand for two different value at the same time
and things could get really confusing!
If you wanted symbols to stand for two different numbers at the same time,
you would need to use two different symbols, like 'x' and 'y'.
So in Algebra, whenever you see the same symbol repeated more than once in an equation,
it's representing the same unknown value.
Like if you see a really complicated Algebraic equation (like this),
where 'x' is repeated a lot of different times,
all of those 'x's stand for the same value,
and it will be your job to figure out what that value is.
Okay, so for any particular equation,
we can't use the same letter to represent two different numbers at the same time,
but what about the other way around?
Could we use two different letters to represent the same number?
Yes! - Here's an example of that.
Let's say you have the equation: a + b = 2
What could 'a' and 'b' stand for so that the equation is true?
Well, If 'a' was 0 and 'b' was 2, then the equation would be true.
Or, we could switch them around.
If 'a' was 2 and 'b' was 0, the equation would also be true.
But there's another possibility:
If 'a' was 1 and 'b' was also 1, that would make the equation true, right?
So, even though 'a' and 'b' are different symbols,
and would usually be used to represent different numbers,
there are times when they might happen to represent the same number.
Oh, and this problem can help us understand something
very important about how symbols are used in Algebra.
Did you notice that there were different possible solution for this equation?
In other words, 'b' could have the value 0, 1, or 2 depending on what the value of 'a' was.
If 'a' is 0 then 'b' must be 2
If 'a' is 1 then 'b' must be 1
If 'a' is 2 then 'b' must be 0
'b' can't have two different values at the same time,
but it's value can change over time if the value of 'a' changes.
In Algebra, 'b' is what's called a "variable" because it's value can 'vary' or change.
In fact, in this equation, both 'a' and 'b' are variables
because their values will change depending on the value of each other.
Actually, it's really common in Algebra to refer to any letter as a variable,
since letters can stand for different values in different problems.
But at Math Antics, we'll usually just use the word "variable"
when we're talking about values that can change or vary in the same problem.
Alright, so far we've learned that Algebra is a lot like arithmetic,
but that in includes unknown values and variables that we can solve for in equations.
There's one other really important thing that I want to teach you
that will help you understand what's going on in a lot of Algebra problems,
and it has to do with multiplication.
Here are the four basic arithmetic operations:
addition, subtraction, multiplication and division.
Although in Algebra, you'll usually see division written in fraction form, like this.
In Arithmetic, all four operations have the same status,
but in Algebra, multiplication get's some special treatment.
In Algebra, multiplication is the 'default' operation.
That means, if no other arithmetic operation is shown between two symbols,
then you can just assume they're being multiplied. The multiplication is 'implied'.
For example, instead of writing 'a' times 'b',
you can leave out the times symbol and just write 'ab'.
Since no operation is shown between these two symbols,
you know that you're supposed to multiply 'a' and 'b'.
Of course, you can't actually multiply 'a' and 'b'
until you figure out what numbers they stand for.
The advantage of this rule about multiplication is that
it makes many algebraic equations less cluttered and easier to write down.
For example, instead of this: a * b + c * d = 10
You could just write: ab + cd = 10.
You can also use this shorthand when you are multiplying
a variable and a known number... like 2x, which means the same thing as 2 times 'x'
or 3y which means the same thing as 3 times 'y'
Since the symbol and the number are right next to each other,
the multiplication is implied.
You don't have to write it down.
Finally some good news!
Now I never have to write down that pesky multiplication symbol again!
Ah, not so fast...
there are some cases in Algebra where still need to use a multiplication symbol.
For example, what if you want to show 2 x 5?
If you just get rid of the times symbol and put the '2' right next to the '5',
it's going to look like the two-digit number twenty-five,
which is NOT the same as 2 x 5.
So, whenever you need to show multiplication between two known numbers,
you still have to use the times symbol, unless...
you use parentheses instead.
But aren't parentheses used to show grouping in math?
How can you use that to show multiplication?
Ah, that's a good question.
Parentheses are used to group things,
but whenever you put two groups right next to each other,
with no operation between them,
guess what operation is implied.
For example, if you see the this,
it means that the group (a+b) is being multiplied by the group (x+y).
We could put a times symbol between the groups, but we don't have to
because it's the default operation in Algebra.
The multiplication is just implied.
So, going back to our problem: 2 x 5
If you wanted to, you could put each of the numbers inside parentheses like this,
and then you could get rid of the multiplication sign.
Now this can't be confused with the number twenty-five,
and since the groups are right next to each other,
you know that you need to multiply them.
Of course, it might seem strange to have just one thing inside 'group' symbols like this,
but it's okay to do that in math.
An alternate way that you could do the same thing would be
to put just one of the numbers in parentheses, like this.
Again, you won't confuse this with a two-digit number
and you know that multiplication is implied.
Okay, so we've learned that Algebra is a lot like arithmetic,
but it involves unknown values or variables that we need to solve for.
And we learned that in Algebra, the multiplication sign is usually not shown,
because it's the default operation.
You can just assume that two things right next to each other are being multiplied.
But why do we even care about Algebra?
Is it good for anything in the 'real world'?
or is it just a bunch of tricky problems that keep students busy in school?
Actually, Algebra is very useful for describing or "modeling" things in the real world.
It's a little hard to see that when you are just looking
at all these numbers and symbols on the page of a math book.
But, it's a lot easier to see
when you start taking algebraic equations and graphing their solutions.
Graphing an equation is like using its different solutions
to draw simple lines and curves that can be used to describe and predict things in real life.
For example, there's a whole class of equations in Algebra
called "linear" equations because they form straight lines when you graph them.
Those sorts of equations could help you describe the slope of a roof
or tell you how long it will take to get somewhere.
Another class of algebraic equations, called "quadratic" equations
can be used to design telescope lenses,
or describe how a ball flies through the air,
or predict the growth of a population.
So Algebra is used all the time in fields like
science, engineering, economics and computer programming.
And even though you might not need Algebra to get by in your day to day life,...
...so divide both sides by 3. That means x = 42.
So... in three... two... one...
Alright... now to see how much butter I need.
It's still a very useful part of math.
Thanks for watching Math Antics, and I'll see ya next time!
Learn more at www.mathantics.com