Alg 1 2.1 Part 1 Write, Interpret, and Simplify Expressions

Summary

This 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

00:00

📘 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.

05:02

🔢 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.

10:03

🧮 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.

15:03

📐 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.

Keywords

💡Expression

An expression is a math phrase made up of numbers, variables, and math operations. It does not have equal sides, which differentiates it from an equation. In the video, expressions are the focus for simplification and interpretation. For example, '16 plus 5' is an expression consisting solely of numbers, while '5a plus 6' includes variables, illustrating the flexibility in what an expression can include.

💡Simplify

Simplifying an expression means to write it in the smallest or most straightforward way possible. The video emphasizes simplifying expressions to their most reduced form, such as changing '16 plus 5' to '21'. This process often involves combining like terms or applying properties like the distributive property.

💡Terms

Terms in an expression are the parts that are added together. They can be constants, variables, or a combination of both. The script mentions '5A plus 6' having two terms, '5A' and '6', which are added together to form the expression. Understanding terms is crucial for simplifying expressions correctly.

💡Coefficient

A coefficient is the numerical part of a variable term. It is just a number that multiplies the variable. For instance, in '6X minus 4Y equals negative 8', the coefficients are '6' for the term '6X' and '8' for '8Y'. Coefficients are essential for understanding how variables are weighted in an expression.

💡Constant

A constant is a term in an expression that does not change regardless of the variable's value. The video uses the number '5' as an example of a constant because it remains the same regardless of what the variable is. Constants are often the non-variable parts of expressions.

💡Distributive Property

The distributive property is a mathematical shortcut for dealing with multiplication and addition. It allows for the multiplication of a number by a sum to be broken down into the multiplication of that number by each addend separately, then adding the results. The video uses '3 times (4 plus 2)' as an example, showing how it can be simplified using the distributive property to '3 times 4 plus 3 times 2', resulting in the same value.

💡Like Terms

Like terms are terms in an expression that have the same variable and exponent parts. They can be combined through addition or subtraction. The video explains that 'X' and '6X' are like terms, as are '3X^2' and '-5X^2', because they share the same variable and exponent. Identifying like terms is necessary for simplifying expressions.

💡Equivalent Expressions

Equivalent expressions are expressions that have the same value but may be written differently. The script mentions '16 plus 5' and '21' as equivalent expressions because they both evaluate to the same numerical value. The concept of equivalent expressions is central to simplifying and evaluating algebraic expressions.

💡Order of Operations

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which mathematical operations should be performed. The video script mentions that order of operations would require adding inside parentheses first, then multiplying by a number outside, unless simplified using the distributive property.

💡Variable

A variable is a symbol, often a letter, that represents an unknown or changing quantity. In expressions, variables can have coefficients and can be part of terms. The video mentions 'X' and 'Y' as examples of variables, which change value and are central to algebraic expressions and equations.

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.