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Hello and welcome to Bay College's video lectures
for Math 085.
This is Section 5.2.
We're going to translate English phrases
to algebraic expressions.
This is how we approach story problems.
So before we begin, my camera operator and I
worked on finding some terminology
that we're going to have to be familiar with.
So Jane and I spent a little bit of time brainstorming,
and maybe we missed a few.
Maybe you can come up with them on your own.
But we're going to find the English words
that we use to describe these mathematical operations.
So first we looked at this one, said, well,
this means addition.
So if we're talking about addition,
that means to "add" or to "plus."
Sometimes we'll see the word "sum" or "combine together."
If we're bringing two things together,
we're going to add them together.
Or maybe we see the word "increase."
So again, that list would be "add" or "addition."
"plus," "sum," "combine together,"
or "increase," sometimes maybe even the word "total."
All right.
When we see this symbol, we know that this is "subtraction."
So that would be one English phrase.
Or we might see "difference."
We might see "take away."
We might see the word "decrease,"
or "less," or "minus."
Any of those English words describe
this mathematical operation.
Now here we have different operators
that indicate the same thing.
Multiplication, we might have "product,"
"times," and the word "of."
So if I have 5 of 6 items, I would have a total of 30 items.
So we look for those words to multiply
or to find the "product," or "times," or "of."
Here we have division, and I showed a fraction here
because that means division and so does the symbol here,
division.
So "division" would be one word. "Quotient" is a frequent word,
or "divide" or "divisible by."
Sometimes even the word "per."
Like "percent" means to divide by 100.
Then when we have exponents, that's this value right here,
we have a base and an exponent.
Some of the English words that describe exponents
are, of course, the word "exponent."
Maybe we use the word "power."
Maybe we use the word "raised to."
And sometimes we have specific words
that specify an actual numerical power, such as "squared."
That means to the second power, or "cubed,"
which means to the third power.
So when we see these terms, they describe
one of these operations.
So let's actually look at some examples
and translate them from the English language
into an algebraic expression.
So the first one says, the product of 3 and x.
So the product, I recognize that to mean multiplication,
the multiplication of 3 and x, which says 3 times x.
And if I want to simplify that I could.
It's just 3x, right, because we know
that this means multiplying.
So does this.
This just means multiplication through adjacency.
All right.
Let's look at the next one, 3 added to a number.
Well, added, I know what mathematical operation that is.
It is that plus sign, right?
3 added to a number.
Well, I don't know what the number is,
so I have to assign a variable.
So I'm going to say 3 plus-- 3 added to a number.
3 added to a number, and that's exactly what that says.
Here's a special word, "twice."
That means a special type of multiplication,
multiplying by 2.
And I recognize that.
That's one we didn't put on our list.
Twice the sum of x and 1.
Here's where we have to be careful because we're not
multiplying 2 times x, or 2 times 1.
It's twice the sum.
We have to multiply 2 times the addition.
So essentially, when I see something like that, it's twice
the sum, I put that sum in parentheses.
The sum of what?
x and 1.
So when I read this, it says twice the sum of x and 1.
All right.
Let's look at this one and see how is this different.
This says twice x plus 1.
And if we think about this in terms of order of operations,
we would essentially multiply before we add, twice x plus 1.
So we identified this, means addition.
This means multiplying by 2.
2x plus 1, twice x plus 1.
And that's what that says.
All right.
Let's look at some more examples.
This one says 5 squared plus the product of x and y.
So 5 squared plus the product of x and y, x times y,
or I can just write it xy.
5 squared plus the product of x and y.
Here we have the quotient-- quotient tells me to divide,
I recognize that-- of a number and 6.
When it comes to division, we have
to identify the numerator and the denominator.
And we usually identify them in that order.
So when I read this, I'd translated in that order.
The quotient of a number-- I'll assign that a variable x--
and 6.
So this is the one that comes first.
This is the one that comes second.
Just like you read from left to right,
you also read one line at a time top to bottom.
So I'm going to write it left to right, top
to bottom, just like I would read.
So the quotient of a number and 6,
the quotient of a number and 6.
All right.
Let's look at the next one.
They get a little bit more complex because we're
integrating more operations.
This one says twice the difference
of a number raised to the third power and 7.
So when I translate this, I know twice means
I'm going to multiply by 2.
Twice the difference.
Well, the difference means subtraction,
but I haven't figured out of what yet.
Well, the difference of a number which is
raised to the third power.
A number raised to a power, I recognize
raised means an exponent and so does power.
So raised to the power, the third power.
Third is 3, so we identified that English word.
Third means 3.
And 7, so this says twice the difference of a number
raised to the third power and 7.
They can get pretty complicated.
But we can hopefully see the point
in this is that if we had to write
every mathematical equation out in English language,
you can imagine how time-consuming that would be.
We have these series of symbols and operators
so that we can make the math a little bit less time-consuming.
All right.
The next one says the quotient of 5
and the sum of a number and 2.
So we have to think of this as like maybe a compound sentence.
The quotient of 5, so the quotient
of 5, because this number came first, and the sum.
Well, and the sum, well, sum tells me to add,
and this would be the next value.
I'm going to have the quotient of 5
and the sum of a number and 2.
A number-- I'll assign that x-- and 2.
So we're able to simplify that.
So what would be good practice is
to take expressions like this and try to write them out
in the English language.
Write it backwards.
If you can do it from this to that,
can you do it from an expression into the English language?
That would be great practice.
And I know you've seen many things
like this in your homework.
So keep practicing.
Do your homework.
And thank you for watching.