An Intro to Combining Like Terms | Simplifying Expressions by Combining Like Terms | Math with Mr. J

Summary

In this video, Mr. J introduces the concept of combining like terms in algebraic expressions. He explains that like terms have the same variables raised to the same power and shows how to simplify expressions by adding or subtracting the coefficients of these terms. Using examples, such as 9x + 3x and more complex expressions, Mr. J guides viewers through the process of identifying and combining like terms to make algebraic expressions simpler and easier to work with. This video is a step-by-step introduction for students learning to streamline their algebraic equations.

Takeaways

• 🧮 Combining like terms helps simplify algebraic expressions.
• 📚 Like terms are terms that have the same variables raised to the same powers.
• ➕ When combining like terms, you only add or subtract the coefficients of the terms.
• 🔢 In expressions without visible exponents, the exponent is understood to be 1.
• ✏️ To simplify expressions, look for terms with the same variables and powers, then combine them.
• 💡 It's helpful to rearrange terms with the same variables next to each other to make combining easier.
• 🔄 You can rewrite expressions with addition instead of subtraction to simplify identification of negative terms.
• ✔️ Example 1: 9x + 3x simplifies to 12x by adding coefficients (9 + 3).
• ✔️ Example 2: 8G + 5G + 7 + 2 simplifies to 13G + 9 by combining G terms and constants.
• 🔀 Rearranging and rewriting expressions can help in organizing terms before combining like terms.

Q & A

What are like terms in algebraic expressions?

Like terms are terms that have the same variables raised to the same powers. Only these terms can be combined in an algebraic expression.

Why is it important to combine like terms?

Combining like terms simplifies an algebraic expression, making it easier to understand and work with while retaining its original value.

In the expression 9x + 3x, why are the terms considered like terms?

Both terms have the same variable, 'x', and the exponent of the variable is the same (in this case, 1), which makes them like terms.

How do you combine like terms in an expression?

To combine like terms, you simply add or subtract the coefficients (the numerical part) of the terms while keeping the variable and its exponent unchanged.

What is the simplified form of the expression 9x + 3x?

The simplified form of 9x + 3x is 12x, since 9 + 3 equals 12, and the variable 'x' remains unchanged.

How would you simplify the expression 8G + 7 + 5G + 2?

First, combine the like terms 8G and 5G to get 13G, and then combine the constants 7 and 2 to get 9. The simplified expression is 13G + 9.

What strategy does the speaker suggest for simplifying more complex expressions?

The speaker suggests rearranging and rewriting the expression so that like terms are placed next to each other. This makes it easier to identify and combine them.

Why is there an 'understood' exponent of 1 in expressions like 9x or 3x?

When there is no exponent written, it is understood that the variable is raised to the power of 1, because any number or variable raised to the power of 1 equals itself.

How can you simplify the expression 6y² + 10y + 2y² + 3y + y?

First, combine the like terms for y²: 6y² + 2y² = 8y². Then combine the like terms for y: 10y + 3y + 1y = 14y. The simplified expression is 8y² + 14y.

What is the advantage of rewriting subtraction as adding the opposite when simplifying expressions?

Rewriting subtraction as adding the opposite helps organize the expression, making it easier to identify negative terms and combine like terms more accurately.

Outlines

00:00

📚 Introduction to Combining Like Terms

In this video, the speaker introduces the concept of combining like terms in algebra. Like terms are defined as terms with the same variables raised to the same power. By combining like terms, algebraic expressions can be simplified, making them easier to work with. The speaker starts with an example, combining the terms 9x and 3x. These terms both contain the variable x raised to the same power, making them like terms. The coefficients (9 and 3) are added, resulting in 12x, which simplifies the original expression without changing its value.

05:03

🔍 Simplifying Expressions by Grouping Like Terms

The second example introduces an algebraic expression with both variable terms and constants: 8G + 7 + 5G + 2. The like terms, 8G and 5G, are identified and combined to get 13G. Similarly, the constants 7 and 2 are added together to get 9. The final simplified expression is 13G + 9. This method reduces the original four-term expression to two terms, illustrating how combining like terms can simplify algebraic expressions efficiently.

📝 Strategies for More Complex Expressions

In this example, the expression 6y² + 10y + 2y² + 3y + y is tackled, which has five terms and includes both squared and linear terms. The speaker emphasizes the importance of rearranging terms to put like terms next to each other. First, the y² terms are grouped (6y² + 2y²), followed by the y terms (10y + 3y + 1y). After rearranging, the coefficients are combined, resulting in the simplified expression 8y² + 14y. This process illustrates how organization can make it easier to combine like terms, especially in more complex expressions.

🧠 Handling Negative Terms in Like Terms

This example features an expression with both positive and negative terms: 7x + 2y - 4x + 2y. The speaker stresses the importance of paying attention to the sign in front of each term. The like terms (7x and -4x) are combined to give 3x, and the like y terms (2y and 2y) are combined to give 4y. The final simplified expression is 3x + 4y. The speaker also demonstrates an alternative strategy of rewriting subtraction as adding the opposite (e.g., changing -4x to + (-4x)) to simplify the process further.

Keywords

💡Like Terms

Like terms are algebraic terms that have the same variables raised to the same power. In the video, 'like terms' are combined to simplify expressions, as seen in the example '9x + 3x' where both terms have the same variable 'x' and are combined to make '12x'.

💡Variable

A variable is a symbol, usually a letter, used to represent an unknown value in an expression or equation. In the video, variables such as 'x', 'y', and 'g' appear in different terms. For instance, in '9x + 3x', 'x' is the variable, and combining the terms simplifies the expression.

💡Coefficient

The coefficient is the numerical factor that multiplies the variable in a term. In the video, the term '9x' has a coefficient of 9. When combining like terms, the coefficients are added together, as in '9 + 3' to give '12x'.

💡Simplify

Simplifying an expression means reducing it to its simplest form by combining like terms. In the video, expressions such as '9x + 3x' are simplified into '12x', making the equation easier to understand and work with.

💡Expression

An expression is a mathematical phrase that can contain numbers, variables, and operators but does not include an equality sign. In the video, expressions such as '8G + 7 + 5G + 2' are discussed, and the goal is to simplify these expressions by combining like terms.

💡Exponent

An exponent is a number that shows how many times a variable is multiplied by itself. In the video, it is mentioned that when there is no visible exponent next to a variable, the exponent is understood to be 1. For example, in 'x' the exponent is implicitly '1'.

💡Constant

A constant is a number that does not change and has no variable attached. In the video, constants such as '7' and '2' are combined separately from the terms with variables, as seen in the expression '8G + 7 + 5G + 2', where the constants '7' and '2' are added to give '9'.

💡Combine

To combine in algebra refers to adding or subtracting like terms to simplify an expression. In the video, the speaker shows how terms like '9x' and '3x' are combined by adding their coefficients to get '12x'.

💡Addition

Addition is the operation of combining two or more numbers or terms. In the video, addition is used to combine like terms, as in '9x + 3x', where the coefficients of the 'x' terms are added to get '12x'.

💡Rearranging Terms

Rearranging terms involves organizing terms so that like terms are next to each other, making it easier to combine them. In the video, this strategy is used for expressions like '6y² + 10y + 2y² + 3y + y', where the speaker rearranges the terms to combine 'y²' terms and 'y' terms separately.

Highlights

Introduction to combining like terms in algebraic expressions.

Definition of like terms: terms with the same variables raised to the same powers.

Simplification of expressions by combining like terms into one term for easier understanding.

Example 1: Combining 9x and 3x by adding coefficients to get 12x.

Explanation of understood exponents of 1 when none are explicitly written.

Example 2: Combining like terms in the expression 8G + 7 + 5G + 2 to get 13G + 9.

Highlighting the concept of constant terms and how they are separate from variable terms.

Example 3: Strategy of rearranging expressions to group like terms, demonstrated with 6y² + 10y + 2y² + 3y + y.

Combining like terms step-by-step in Example 3 to simplify to 8y² + 14y.

Example 4: Combining 7x and -4x, and 2y and 2y, resulting in 3x + 4y.

Emphasis on carrying the sign of terms when rewriting expressions, especially for negative terms.

Second strategy introduced: rewriting expressions by changing subtraction to adding the opposite.

Rewriting subtraction as adding the opposite helps clarify negative terms.

Simplification of expressions demonstrated in multiple ways to offer alternative strategies for combining like terms.

Final simplified expressions in all examples are equivalent to the original expressions, maintaining their value.

Transcripts

[Music]

welcome to math with Mr

J in this video I'm going to go through

an introduction to combining like terms

now remember like terms are terms with

the same variables to the same Powers

when we combine like terms we look for

any like terms in the given algebraic

expression and combine them into one

term by combining like terms we can

simplify expressions that just means we

can rewrite the original expression in a

simpler and easier way to understand and

work with let's jump into number one

where we have 9x + 3x we will start with

this basic expression and work our way

up so we have two terms in this

expression 9x and 3X both terms have the

same variable of X and these variables

of X are to the same power remember when

we don't have an an exponent attached to

a variable there is an understood

exponent of one anything to the power of

one is just itself so 9x and 3X are like

terms now when we combine like terms all

we need to do is add or subtract the

coefficients the numbers in front of the

variables the coefficients in number one

are 9 and 3 we have a positive 9x plus a

positive 3x so let's let's add those

coefficients 9 + 3 is 12 and then we

have the variable of X and that's it we

took those two like terms 9x and 3X and

combined them into one term 12x 12x is

equivalent to 9x + 3x so we didn't

change the value of the expression so

12x is our final simplified expression

let's move on to number two where we

have 8 G + 7 + 5G + 2 are there any like

terms that we can combine in order to

simplify this expression yes we have

8G and 5G both of those terms have that

variable of G and then we have constant

terms

7 and 2 I'll box in the constant terms

to separate them from The 8G and the 5G

now we can combine like terms we have 8G

+ 5G that gives us

13g and then we have 7 + 2 that gives us

9 so we end up with 13 G + 9 and that's

our simplified expression that

expression of 13 G + 9 is equivalent to

the original expression we were just

able to simplify the original expression

by combining like terms we started with

four total terms but we were able to

combine like terms and now we only have

two total terms let's move on to number

three where we have 6 y^ 2 + 10 y + 2

y^2 + 3 y + y let's find any like terms

that we can combine we'll start with 6

y² 2 y^2 is a like term both of those

terms have that variable of Y to the^ of

2 now do we have any other like terms

within this algebraic expression that we

can combine yes 10 Y and I will box

these terms in in order to separate them

from the Y squar terms 3

Y and then y now I do want to mention

this term right here the Y the variable

by itself the coefficient is one we

don't have a coefficient written in

front whenever you see that the

coefficient is one there is an understod

one in front of a variable and it can be

helpful to write that one in there when

you combine like terms so you can always

write that one if you would like now

since this algebraic expression has five

terms and we are working our way up to

more complicated algebraic expressions

we're going to use a strategy to help

help us organize the expression before

we combine like terms we are going to

rearrange and rewrite the expression and

put the like terms next to each other

I'll start with 6

y² plus the like term of 2

y^2 plus now we have the Y terms so 10 y

+ 3 y + 1 y so now all of the like terms

are next to each other and it's a little

easier to see what we can combine so

this is a strategy to keep in mind now

do you have to do this step in order to

combine like terms no but it can be

helpful now we can combine like terms we

will start with 6 y^2 + 2 y^ 2 so add

the coefficients 6 +

2 is 8

and then we have

y^2

plus now we can combine the Y terms so

we have 10 + 3 + 1 10 + 3 is 13 + 1 is

14 so we get 8 y^2 + 14 Y and that's the

simplified expression we now have an

equivalent expression that is simpler

than the original We simplified the

expression we went from five terms to

two terms let's move on to number four

where we have 7x + 2 y - 4x + 2 y let's

find any like terms that we can combine

we will start with 7x and

-4x now when we combine like terms a

term is going to take the sign that's in

front of it so this is -4x

then we have 2 Y and 2 y so let's box

those terms in in order to separate them

from the X terms now we can rewrite this

expression with the like terms next to

each other we will start with

7X - 4x so we have a -4x there make sure

to bring the sign that's in front of the

term with it when we rewrite the

expression Plus

2 y so now we have the Y terms plus

another 2 y now we can combine like

terms so we have 7X - 4x or you can

think of this as 7x being combined with

-4x however you want to think about it 7

- 4 is 3 and then we have the x or if

you're thinking about it as 7x combined

with a NE 4X 7 and -4 give us 3 as well

then we have our 2 y + 2 y that gives us

plus 4 y so we end up with

3x+ 4 Y and that's our simplified

expression we went from four total terms

to two total terms by combining like

terms 3x + 4 Y is equivalent to the

original expression we were just able to

again simplify this expression by

combining like terms now I also want to

go through simplifying this expression a

slightly different way to start off and

that's by rewriting the original

expression with only addition separating

the terms we do this by changing any

subtraction to adding the opposite the

benefit of having all terms separated

only by addition is that it's a little

simpler to identify all of the terms

especially any negative terms it kind of

organizes the expression and helps any

negatives stand out I'll rewrite the

expression off to the side here so 7x +

2 y -

4x + 2 y so let's rewrite subtraction as

adding the opposite so adding the

opposite of a positive 4X is is

a 4X so adding the opposite let's

rewrite the expression with that change

so we have 7 x + 2 y +

-4x + 2 y let's rewrite that expression

with like terms next to each other so

7x

+

-4x + 2 y + 2 y now we can combine like

terms we have 7 x + -4x that gives us

3x and then we have 2 y + 2 y so that

gives us plus 4 y 3x + 4 y that way as

well so that's just another strategy to

be aware of so there you have it there's

an introduction to combining like terms

I hope that helped thanks so much for

watching until next time peace