An Intro to Combining Like Terms | Simplifying Expressions by Combining Like Terms | Math with Mr. J
Summary
TLDRIn 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
📚 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.
🔍 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.
Mindmap
Keywords
💡Like Terms
💡Variable
💡Coefficient
💡Simplify
💡Expression
💡Exponent
💡Constant
💡Combine
💡Addition
💡Rearranging Terms
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
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