Simplifying Algebraic
Expressions
Mac „N Cheese Math
Remember Algebraic Expressions?
• An Algebraic Expression-a group of numbers, symbols,
and variables that express an operation or a series of operations.
• Anatomy of an Algebraic Expression:
2x + 7
coefficient
•
variable
constant
the number in front
a number without a variable
of the variable
Each separate piece of the expression, separated by + or - is called
a term. 2x is a term and 7 is a term.
Imagine you‟re at the grocery store,
and you LOVE mac „n cheese…..
• You buy one box of
brand X mac „n
cheese
• That makes one X
X
• You buy three boxes
of brand X mac „n
cheese
X
X
X
• That makes 3 X‟s
As an algebraic
expression, this would
be 3X
(a box of mac „n
cheese times 3)
How would you write as an
algebraic expression…
• 4 boxes of brand X
• 4x
mac „n cheese?
• 8 boxes of mac „n
• 8x
cheese?
• What if you bought 4
• 4x + 5x = 9x
boxes for yourself and
Your 4 boxes plus her
5 boxes for your
5 boxes equals 9
sister?
boxes
Now, let‟s say that there‟s also a jumbo size
box of the same mac „n cheese available.
We‟ll call this size X2
• You buy one box of the
jumbo size.
• As an algebraic
expression, this would be
X2 (one X2)
X2
• You buy 4 jumbo boxes
for yourself and 3 jumbo
boxes for your sister
• 4X2 + 3X2 = 7X2
Your 4 jumbos, plus
your sister‟s 3
jumbos, equals 7 total
jumbos
What if you only wanted 2 regular size
boxes and your sister wanted 4 jumbo size
boxes?
You buy two regular boxes
2X (2 regular sizes)
Plus your sister‟s
4X2 (4 jumbo boxes)
Do you say, “I just bought 6 regular-jumbo
boxes”
or..
Do you say, “I just bought 2 regular boxes plus 4 jumbo
boxes”
Algebraic expressions work the
same way.
•
•
•
You can‟t add or subtract two terms, unless they have
the same variable with the same exponent.
Two terms with the same variable and the same
exponent are called like terms
For example, you can add …
1.
2.
•
2x + 3x = 5x b/c they have the same variable with the same
exponent (remember both variables have a 1 that is
understood)
2y2 + 5y2 = 7y2 They are like terms!
But you can‟t add…
1. 2y + 2y2
2. 5x + 5x4
Because they don‟t have the same variable AND exponent.
• Constants (numbers without variables) are
only like terms with other constants
Example # 1
• Simplify 3x + 6x
•
x
• (3+6)x
• 9x
• Simplify by
combining (adding)
like terms
• Variable stays the
same
• Add the coefficients
• Final Answer
Example # 2
• Simplify 2x + 5 + 6x
• 2x + 6x + 5
• (2+6)x + 5
• 8x + 5
• 2x and 6x are like
terms
• Commutative
Property
• Combine Like Terms
• Simplified; 8x and 5
are not like terms, so
you are finished
Example # 3
• Simplify
3x – 5x
• 3x + (-5x)
• (3+-5)x
• -2x
•
Definition of subtraction
(change your minus to a plus,
change the sign of next)
• Combine like terms
(variable stays the same, add
coefficients)
• answer
Example # 4
• Simplify -4x – 6x +5
• -4x + (-6x) + 5
•
Change your minus to a plus,
change the sign of next
(definition of subtraction)
•
Add like terms; variable stays
the same
•
10 x and 5 are not like terms,
so you are finished
• (-4+-6)x + 5
• -10x + 5
When there is no coefficient in front of the variable, there is
actually a one in front of it that we don‟t write. Just like in
exponents, the one is understood
• Examples:
3x + x (3 x‟s plus one
more)
(3 + 1)x
4x
• 7x – x (7 x‟s take
away one)
(7-1) x
6x
Identify the terms, like terms, coefficients, and constants in the
expression
Definition of
subtraction
Identity
Property
Answer: The terms are 4x, –x, 2y, and –3. The like terms are 4x and
–x. The coefficients are 4, –1,
and 2. The constant is –3.
Identify the terms, like terms, coefficients, and constants in the
expression
Answer: The terms are 5x, 3y, –2y, and 6. The like terms are 3y and
–2y. The coefficients are 5, 3,
and –2. The constant is 6.
Simplify
.
5x and 4x are like terms.
Distributive Property
Simplify.
Answer: 9x
Simplify
.
8n and 4n are like terms.
Commutative Property
Distributive Property
Simplify.
Answer:
Simplify
.
6x and –5x are like terms. 4 and –7 are also like terms.
Definition of
subtraction
Commutative
Property
Distributive
Property
Simplify.
Answer:
Sometimes you have some extra
work to do before you can combine
like terms.
• m + 3(m + 2)
When you have a
number outside of
addition or subtraction
in parenthesis, you
should first use
Distributive Property
• = m + 3∙m + 3∙2
• = m + 3m + 6
• = 4m + 6
•
Distribute the 3 inside the
parenthesis
•
Combine like terms to simplify
Sometimes your distributive property gets more
complicated with subtraction signs
• 3x – 4(x + 3)
• 3x + (-4)(x + 3)
•
•
•
•
3x + (-4)(x) + (-4)(3)
3x + -4x + -12
-x + -12
-x - 12
•
You don‟t have to, but in a
case like this, it is easier to
change the subtraction to
addition and then change the
sign of the number behind the
subtraction sign
•
Use distributive property
•
Combine like terms
•
Make it a little more simple by
getting rid of the + -
Sometimes you have to use the distributive property to get rid of
parenthesis before you can combine like terms.
Distributive
Property
Multiply.
Identity Property
Commutative
Property
Distributive
Property
Simplify.
Answer:
Simplify each expression.
a.
Answer:
b.
Answer:
c.
d.
Answer:
Answer:
Work You and a friend worked in the school store last week. You worked
4 hours more than your friend. Write an expression in simplest form that
represents the total number of hours you both worked.
Words
Your friend worked some hours. You worked
than your friend.
Variables
Let
Let
Expression
4 more hours
number of hours your friend worked.
number of hours you worked.
To find the total, add the expressions.
Associative Property
Identity Property
Distributive Property
Simplify.
Answer: The expression
represents the total number of
hours worked, where h is the number of hours your friend worked.
Library Books You and a friend went to the library. Your friend borrowed
three more books than you did. Write an expression in simplest form that
represents the total number of books you both borrowed.
Answer:
Find
.
Method 1 Add vertically.
Align like terms.
Add.
Method 2 Add horizontally.
Associative and
Commutative
Properties
Answer: The sum is
10w + 1.
Find
.
Method 1 Add vertically.
Align like terms.
Add.
Method 2 Add horizontally.
Write the expression.
Group like terms.
Simplify.
Answer: The sum is
Find
.
Write the expression.
Simplify.
Answer: The sum is
Find
.
Leave a space because there is no
other term like xy.
Answer: The sum is
.
Find each sum.
a.
b.
Answer:
c.
Answer:
d.
Answer:
Answer:
Geometry The length of a rectangle is
units and the width is 8x
– 1 units.
Find the perimeter.
Formula for the perimeter of a
rectangle
Replace
with
and w with
Distributive Property
Group like terms.
Simplify.
Answer: The perimeter is
Find the length of the rectangle if
Write the expression.
Replace x with –3.
Simplify.
Answer: The length of the rectangle is 16 units.
Geometry The length of a rectangle is
units and the width is 6w
a. Find the perimeter.
Answer:
b. Find the length if
Answer:
39 units
– 3 units.
Find
.
Align like terms.
Subtract. The subtraction sign is distributed to everything
inside of the parenthesis behind it, so that this is -9a and
-2.
Answer: The difference is
.
Find
.
Align like terms.
Subtract. Remember to distribute
the subtraction sign to everything
in parenthesis behind it!
Answer: The difference is
.
Find each difference.
a.
Answer:
b.
Answer:
Find
.
To subtract (3x +
add (–3x – 9).
9),
Group the like terms.
Simplify.
Answer: The difference is x–17.
Find
.
The additive inverse of
Align the like terms and add the additive inverse.
Answer:
Find each difference.
a.
Answer: 10c
b.
Answer:
– 7.
Geometry The length of a rectangle is
units. The width is
units. How much longer is the length
than the width?
difference in measurement
Substitution
Add additive inverse.
Group like terms.
Simplify.
Answer:
The length is
than the width.
units longer
Profit The ABC Company’s costs are given by
where x = the number of items produced. The revenue is
given by 5x. Find the profit, which is the difference between the
revenue and the cost.
Answer:
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