Alg 1 2.1 Part 1 Write, Interpret, and Simplify Expressions
Summary
TLDRThis educational video script focuses on algebraic expressions, emphasizing the importance of understanding their components and simplification. It defines expressions as math phrases consisting of numbers, variables, and operations, distinct from equations due to the absence of equal signs. The script introduces terms, coefficients, and constants, highlighting their roles in expressions. It also explains the distributive property as a shortcut for multiplication over addition and subtraction, and illustrates how to identify and combine like terms to simplify expressions. Practical examples are provided to clarify these concepts.
Takeaways
- 📘 An expression is a math phrase made up of numbers, variables, and operations without an equal sign.
- 🔢 Expressions can be simplified to their smallest form, like reducing 16 + 5 to 21.
- 🔑 The terms of an expression are the parts that are added together, like 5A and 6 in the expression 5A + 6.
- 🆗 To simplify expressions, rewrite subtractions as additions of a negative to make it easier to identify like terms.
- 📐 Coefficients are the numerical parts of variable terms, like the 6 in 6X or the 8 in 8Y.
- 👉 If a variable stands alone without a number, its coefficient is understood to be 1.
- 🔄 The distributive property allows multiplying a number by a sum, like a * (b + c), by multiplying the number by each term inside the parentheses and then adding the results.
- 📚 Like terms are terms with the same variable and exponent parts, such as X and 6X or 3X^2 and -5X^2.
- 📉 Constants are terms without variables, like the number 5, which remain unchanged regardless of variable values.
- 📌 Equivalent expressions are different ways of writing the same value, such as 16 + 5 and 21.
Q & A
What is an expression?
-An expression is a math phrase made up of numbers, variables, and math operations. It does not have equal sides and is not an equation.
What is the difference between an expression and an equation?
-An expression is a math phrase without equal signs, while an equation has an equal sign and represents a statement that two expressions are equal.
What are equivalent expressions?
-Equivalent expressions are expressions that have the same value but may be written in different ways, such as 16 plus 5 and 21.
What are the terms in an expression?
-The terms in an expression are the parts that are added together. For example, in the expression 5A plus 6, 5A and 6 are the terms.
How do you rewrite an expression to show addition when there is subtraction?
-You rewrite subtraction as adding a negative. For instance, 6x minus 4 plus 8y can be rewritten as 6x plus negative 4 plus 8y to show addition.
What is a coefficient in an algebraic expression?
-A coefficient is the numerical part or the number part of a variable term in an expression. For example, in the term 6X, the coefficient is 6.
If a variable term has no number part, what is its coefficient?
-If a variable term has no number part, like X plus five, the coefficient is one, as it implies one X.
What is a constant in an algebraic expression?
-A constant is a term in an expression that does not change regardless of the variables' values, such as the number 5 in an expression.
How does the distributive property simplify expressions?
-The distributive property allows you to multiply a number by each term inside a parenthesis and then add the results, simplifying the expression, like 3 times (4 plus 2) can be simplified to 3 times 4 plus 3 times 2.
What are like terms?
-Like terms are terms in an expression that have the same variable and exponent parts, such as X and 6X, or 3X squared and negative 5X squared.
Why can you only add like terms?
-You can only add like terms because they have the same variable and exponent parts, allowing them to be combined into a single term.
Outlines
📘 Understanding Algebraic Expressions
This section introduces the concept of algebraic expressions, which are math phrases composed of numbers, variables, and operations without equal signs. The script explains that expressions can be simplified to their smallest form, such as rewriting '16 plus 5' as '21'. It also distinguishes between terms within an expression, which are parts that are added together, and coefficients, which are the numerical parts of variable terms. An example is given where '6x minus 4y equals negative 8', where '6' and '8' are coefficients. The script clarifies that a term without a numerical coefficient, like 'x plus five', has a coefficient of one by default. The importance of rewriting expressions to show addition, such as changing '6x minus 4 plus 8y' to '6x plus negative 4 plus 8y', is also discussed.
🔢 Exploring Coefficients and Constants
The script delves into the difference between coefficients and constants. Coefficients are the numerical parts of variable terms, while constants are numbers that do not change regardless of variable values. For instance, in '2x plus five', '2' is a coefficient and '5' is a constant. The video then reviews the distributive property, which allows for the simplification of expressions involving multiplication and addition. An example is given with '3 times (4 plus 2)', which can be simplified using the distributive property to '3 times 4 plus 3 times 2', resulting in '18'. The concept of like terms is introduced, which are terms with the same variable and exponent parts, and only like terms can be added together to form equivalent expressions.
🧮 Simplifying Expressions Using Distributive Property
The script provides examples of simplifying algebraic expressions by identifying like terms and applying the distributive property. It emphasizes the importance of rewriting subtraction as addition of a negative to avoid confusion with signs. The process involves distributing multiplication across terms and then combining like terms. An example is given where '5x plus one' cannot be simplified further since they are not like terms, but '4 times (5x plus 1 minus 3x)' can be simplified using the distributive property to '20x plus 4', and then further simplified by combining like terms to '17x plus 4'. The script also demonstrates how to handle multiple negative signs and parentheses in expressions.
📐 Applying Order of Operations and Simplifying
This section focuses on the application of the order of operations in simplifying algebraic expressions, particularly the handling of parentheses and multiplication before addition. The script shows how to distribute multiplication across terms inside parentheses and then combine like terms. An example is given where 'two-thirds times (3x plus nine minus 4 minus 7 times 21x minus 14)' is simplified by first distributing the multiplication and then combining like terms to get 'negative 10x plus 14'. The use of a calculator for handling fractions is also mentioned, and the script advises rewriting subtractions as additions of negatives to simplify the process.
Mindmap
Keywords
💡Expression
💡Simplify
💡Terms
💡Coefficient
💡Constant
💡Distributive Property
💡Like Terms
💡Equivalent Expressions
💡Order of Operations
💡Variable
Highlights
Definition of an expression as a math phrase made up of numbers, variables, and operations.
Explanation that expressions do not have equal signs and are not equations.
Simplification of expressions to their smallest possible form.
Introduction to terms within an expression as parts that are added together.
Rewriting expressions to show addition by converting subtraction to adding a negative.
Definition of a coefficient as the numerical part of a variable term.
Explanation that a variable without a number has a coefficient of one.
Differentiation between coefficients and constants, with constants not changing regardless of variable values.
Overview of the distributive property as a shortcut for multiplication and addition.
Illustration of the distributive property using numerical examples.
Identification of like terms based on having the same variable and exponent parts.
Explanation that only like terms can be added together.
Process of simplifying expressions by identifying and using like terms.
Use of the distributive property to simplify expressions involving parentheses.
Technique of rewriting subtraction as addition of a negative to simplify expressions.
Step-by-step example of simplifying an expression using the distributive property and combining like terms.
Emphasis on not using equal signs when simplifying expressions.
Advice on using a calculator for dealing with fractions during simplification.
Transcripts
section 2.1 we're going to write
interpret and simplify expression so our
objective is to write an algebraic
expression interpret the parts of it and
Use the distributive property to
simplify it
so first vocab we need to know what an
expression is
an expression
is a math phrase
it is made up
of numbers
variables
and math operations
okay so for example a expression
16 plus 5.
is an expression this one's just made up
of numbers
um
but it's just a phrase
um 5 a plus six
is also a phrase one thing to note
Expressions
do not
have equal sides
if it's an equal sign
it's an equation not an expression
Expressions do not have equal signs
okay so it's just a phrase now we can
simplify those phrases so like for
example the phrase 16 plus 5 I could
simplify that to 21. okay these are what
we call equivalent expressions
they have the same value
but they're written in different ways so
often they'll you'll be asked to
evaluate an expression
or simplify
an expression and really what that means
is to get it written in the smallest way
possible 16 plus five
can be written smaller it can be written
as 21 okay
all right so the different parts of an
expression we call its terms the terms
are the parts that are added together
foreign expression
okay so for example this one that I did
up here
this 5A plus 6 has two terms a 5A and a
six those two pieces are added together
okay it's got two terms
um sometimes you might have to rewrite
an expression to show that it's addition
what I mean by that if I had 6x
minus 4 Plus 8y well I have a
subtraction in here so the first thing
I'm going to do is I'm going to rewrite
that subtract as adding a negative
so there I can see the three terms a 6X
and negative 4 and an 8y it has three
terms
a 6X a negative 4 and a 8y
now a coefficient
a coefficient is the numerical part or
the number part
of a variable term
coefficient is just a number
okay so in this previous problem up here
the 6X minus 4y equals negative 8. the
number part in the variable term so this
first term is a variable term it's got
an X in it so the coefficient
is just the 6.
this 8y
is a variable term so the coefficient is
just an eight
okay well and we've had the discussion
before if I just have a variable
with no number part for example let's
take X plus five
that has a coefficient of one because
it's only one X so the coefficient is
one if there isn't a number written in
front of it
one just tends to be that number we
don't write okay it's only one X as
opposed to this one having
six X's
okay
now your turn
I want you to go ahead pause the video
and I want you to answer these three
questions how many terms does that have
what are the coefficients and is five a
coefficient
okay all right now that you've done that
your turn problem I want to talk about
that number five
okay that number five is not a variable
term
okay so do we call it a coefficient
no coefficients so this last one is five
a coefficient no
it's actually what we called it is
called
a constant
because it does not change that 5 is
always five regardless of what the
variable is
as the variable changes here as X
changes this 2x term is going to change
as y changes this negative 8y to the
second term is going to change but this
5 doesn't care what x is doesn't care
what Y is it's always five hence why we
call it a constant so that's some
important vocabulary there
it's called a constant
okay all right now a review of the
distributive property the distributive
property is a way to say shortcut a
multiplication and addition so basically
what this distributive property tells me
in symbols if I have a
times B plus c
okay so here I have a order of
operations would tell me I would need to
add B plus C first
in parentheses and then multiply by a
but sometimes that's not possible
especially when if B and C are variables
so with the distributive property tells
me is that I can multiply the a times
the B
multiply the a
times the C
and then add them and it's add because
this middle term is ADD
okay
so that is the distributive property how
that would look in numbers if I'm taking
three
times four plus two
I could follow order of operations and
add four plus two
and we're going to do it both ways or I
could take 3 times the 4.
take 3 times the 2 and add it
3 times 4 is 12. plus 3 times 2 is 6 and
I get 18. that would give me the exact
same thing if I take 3 times 4 plus 2 is
6 3 times 6 is also eighteen
they are equivalent expressions
using the distributive property
okay
like terms when it comes to simplifying
these Expressions we have to be able to
identify which terms are like or alike
like terms are any terms whose variable
and their exponents are the same
variable and exponent
parts
are the same
okay so the term X and 6x
are like and notice I don't use an equal
sign they're not equal they're just
alike whereas x to the second
let's go 3x to the second
and negative 5x to the second
are also like because they're x to the
seconds
but like an X term
um 6 y to the Third
okay and negative one-half
y to the third put a comma in between
are alike because they're both y to the
third the entire entire variable part
has to be the same
x squared y x to the second y
and two-thirds x to the second y are
alike but notice X is to the second in
both of them and they both have a y
ay
they're like terms
okay again the X has to be to the second
and they each have to have a y
um not like terms of this if I had X Y
to the second
that's not a like let's put a three in
front this one is not a like to that one
because the variable part's different
the exponent part's different
okay
so we're going to be spending a lot of
this next section looking at like terms
because you can only add like terms
okay because we're going to be writing
what's called equivalent expressions
equivalent expressions are expressions
that have the same value
I talked about that earlier when I wrote
16 plus 5. 16 plus 5 and 21
are equivalent expressions okay
I want you to go ahead and pause the
video and add um do this next your turn
problem which of those following are
like terms
all right now that you've done that your
turn problem let's go ahead and let's do
some examples we're going to simplify
each expression
here's the Expressions we're simplifying
so this is going to involve us
identifying like terms distributive
property so forth and so on okay so the
first thing I notice order of operations
tells me I really should do inside
parentheses first
but these two terms are not alike 5x
plus one
they're not alike I can't add them so I
can't do inside parentheses first so I
have my parentheses multiplied by 4.
here I have my multiply by four that's a
multiply I put my DOT in there so here
I'm going to use the distributive
property I'm going to take 4
times each of those pieces so I'm going
to take 4 times 5x which is
20x and then 4 times 1
Plus 4.
and then I'm going to rewrite
this next bit
this subtract 3 as plus a negative 3x
so now it's straight across addition
so since it's straight across addition I
can add in any order so I'm going to add
my like terms these two are like they
both have X's
so I can add that they're both X's
they're alike I can add so 20x plus
negative 3x so 20 basically I always
think okay 20 cats negative three cats
I'm going to have a total of how many
cats they're still cats they're still
going to be X's so same signs add and
keep different signs subtract so that's
going to give me 17 X's
and then I haven't done anything with
that plus four so he comes along notice
how I'm not using any equal signs I'm
just working down my page one line then
the next
hey um and then I look do I have any
more like terms in here this is an X
term this is not nope they're not alike
so there's as far as I can go with it
each of these are equivalent expressions
this is equivalent to this
and this is equivalent to this okay
we're just rewriting it
hey holy negative sign for this next one
there are so many negatives in here
um I'm going to start by rewriting this
and changing all of my subtracts to plus
a negative trust me it makes life easier
so this first one is going to subtract
that's a negative seven
this next one x minus 5 I'm going to
write it as X plus a negative 5.
now this next one is subtract 3 plus a
negative 3.
then 4 plus a negative 5X
I can almost guarantee if you don't do
this step you're going to get your signs
wrong your negatives are going to get
all mixed up
Okay order of operations work inside
parentheses first can I nope not like
terms X and a negative 5 can't add them
so distribute it
distribute
so I'm going to take negative 7 times x
is negative 7X
plus negative 7 times negative 5 is a
positive 35.
a next bit I cannot do this addition
until I deal with the parentheses or
multiplication order of operations I
cannot add until I parentheses and
multiply
so I count parentheses because they're
not like terms
but I can multiply
so I'm going to go ahead and I have to
distribute here as well so Plus
from this Plus
and we have negative 3 times 4 negative
12.
plus that's really from this one
negative 3 times negative 5x is negative
3 times negative 5 is positive 15x
now identify our like terms here I have
an X term
everything is adding add add add so I
can add in any order so let's go ahead
and let's add it to my other my 15. so
negative 7 plus 15 same signs add and
keep different signs subtract so that
gives me 8x
now I have my constants
my 35 and my negative 12 which is going
to give me a
Plus what's 35 plus negative 12 that's
going to be a what 23
okay they are no longer like terms 8x23
can't add them
okay next
this one doesn't have nearly as many
negatives but it still has a couple
so the first thing I'm going to do is
I'm going to rewrite those subtracts
as negatives like we did in that last
step so two-thirds please use two lines
of paper for your fractions
three X plus nine
now I'm going to rewrite the subtract as
plus a negative plus a negative
four sevens
21x plus a negative 14. trust me you're
going to save yourself a lot of
Heartache by doing that first now I need
to distribute well first look inside my
parentheses here
I can't put these together one's the
next one's a nine if you can put by all
means put them together so I need to
take two-thirds times three now I could
spend a lot of time coming over here to
the side and taking two thirds times
three that's actually not that bad
um
two-thirds times three over one gives me
what six thirds which is two so it ends
up being two x because there's still an
X there but if that makes you nervous we
can always pull up our calculator in
parentheses Take 2 divided by three
that's two-thirds times three and it
still gives me two okay so your
calculator is completely okay to deal
with those fractions I need to take
two-thirds times nine
calculator parentheses two-thirds times
nine
is six
so plus six
plus now I need to take
distribute because again I can't add
these in here one's got an x on it
negative 4 7 times 21
a negative 4 7 times 21 and again with a
graphing calculator I don't need to like
clear out the previous stuff I can just
go ahead and leave it up there negative
12.
so negative negative 12x
and then take negative 4 7 times
negative 14.
negative 4 7 times negative fourteen is
eight
plus eight now again everything's
addition
so I can go ahead and add my like terms
so 2x negative 12. 2x plus negative 12
is negative 10.
plus six and the 8 gives me 14.
negative 10x Plus 14.
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