Download 1.1 and 1.2 - David Beydler`s Math

Math 71A
1.1 – Algebraic Expressions and Real
Numbers
1.2 – Operations with Real Numbers
and Simplifying Algebraic Expressions
1
Order of Operations
1. Parentheses (in general, grouping symbols:
( ), [ ], { } ,
,
)
2. Exponents (and roots and absolute values)
3. Multiplication/division (from left to right)
4. Addition/subtraction (from left to right)
2
ex: 2 1 + 3 2 + 1
+ 9 + 16 =
3
ex: Evaluate 7 + 5 𝑥 − 4
3
for 𝑥 = 6.
4
A letter used to represent various numbers is
called a _________________.
ex:
A letter used to represent a particular number is
called a _________________.
ex:
5
A letter used to represent various numbers is
variable
called a _________________.
ex:
A letter used to represent a particular number is
called a _________________.
ex:
6
A letter used to represent various numbers is
variable
called a _________________.
ex: 𝒙, 𝒚, 𝑪, 𝒕
A letter used to represent a particular number is
called a _________________.
ex:
7
A letter used to represent various numbers is
variable
called a _________________.
ex: 𝒙, 𝒚, 𝑪, 𝒕
A letter used to represent a particular number is
constant
called a _________________.
ex:
8
A letter used to represent various numbers is
variable
called a _________________.
ex: 𝒙, 𝒚, 𝑪, 𝒕
A letter used to represent a particular number is
constant
called a _________________.
ex: 𝒎 = # of months in a year
9
A letter used to represent various numbers is
variable
called a _________________.
ex: 𝒙, 𝒚, 𝑪, 𝒕
A letter used to represent a particular number is
constant
called a _________________.
ex: 𝒎 = # of months in a year = 𝟏𝟐
10
A letter used to represent various numbers is
variable
called a _________________.
ex: 𝒙, 𝒚, 𝑪, 𝒕
A letter used to represent a particular number is
constant
called a _________________.
ex: 𝒎 = # of months in a year = 𝟏𝟐
𝝅 ≈ 𝟑. 𝟏𝟒,
𝒆 ≈ 𝟐. 𝟕𝟏𝟖𝟐𝟖
11
The combination of variables, constants, numbers,
+, −, ⋅, ÷, powers, roots, etc. is called an
_____________________________.
ex:
Two algebraic expressions with an equal sign in
between is called an ________________________.
ex:
12
The combination of variables, constants, numbers,
+, −, ⋅, ÷, powers, roots, etc. is called an
algebraic expression
_____________________________.
ex:
Two algebraic expressions with an equal sign in
between is called an ________________________.
ex:
13
The combination of variables, constants, numbers,
+, −, ⋅, ÷, powers, roots, etc. is called an
algebraic expression
_____________________________.
𝟐𝒙
ex:
𝒙 + 𝟏,
, 𝟐( 𝒙 − 𝟏)
𝟑
Two algebraic expressions with an equal sign in
between is called an ________________________.
ex:
14
The combination of variables, constants, numbers,
+, −, ⋅, ÷, powers, roots, etc. is called an
algebraic expression
_____________________________.
𝟐𝒙
ex:
𝒙 + 𝟏,
, 𝟐( 𝒙 − 𝟏)
𝟑
Two algebraic expressions with an equal sign in
equation
between is called an ________________________.
ex:
15
The combination of variables, constants, numbers,
+, −, ⋅, ÷, powers, roots, etc. is called an
algebraic expression
_____________________________.
𝟐𝒙
ex:
𝒙 + 𝟏,
, 𝟐( 𝒙 − 𝟏)
𝟑
Two algebraic expressions with an equal sign in
equation
between is called an ________________________.
ex:
𝟓
𝑪 = 𝑭 − 𝟑𝟐
𝟗
16
The combination of variables, constants, numbers,
+, −, ⋅, ÷, powers, roots, etc. is called an
algebraic expression
_____________________________.
𝟐𝒙
ex:
𝒙 + 𝟏,
, 𝟐( 𝒙 − 𝟏)
𝟑
Two algebraic expressions with an equal sign in
equation
between is called an ________________________.
ex:
𝟓
𝑪 = 𝑭 − 𝟑𝟐
𝟗
𝑬 = 𝒎𝒄𝟐
17
The combination of variables, constants, numbers,
+, −, ⋅, ÷, powers, roots, etc. is called an
algebraic expression
_____________________________.
𝟐𝒙
ex:
𝒙 + 𝟏,
, 𝟐( 𝒙 − 𝟏)
𝟑
Two algebraic expressions with an equal sign in
equation
between is called an ________________________.
ex:
𝟓
𝑪 = 𝑭 − 𝟑𝟐
𝟐
𝟐𝒙
+ 𝟔𝒙 = 𝟎
𝟗
𝑬 = 𝒎𝒄𝟐
18
Absolute Value
The absolute value bars make the number inside
positive.
ex: −4 = 𝟒
3 =𝟑
0 =𝟎
−𝜋 = 𝝅
19
Absolute Value
The absolute value bars make the number inside
positive.
ex: −4 = 𝟒
3 =𝟑
0 =𝟎
−𝜋 = 𝝅
20
Absolute Value
The absolute value bars make the number inside
positive.
ex: −4 = 𝟒
3 =𝟑
0 =𝟎
−𝜋 = 𝝅
21
Absolute Value
The absolute value bars make the number inside
positive.
ex: −4 = 𝟒
3 =𝟑
0 =𝟎
−𝜋 = 𝝅
22
Absolute Value
The absolute value bars make the number inside
positive.
ex: −4 = 𝟒
3 =𝟑
0 =𝟎
−𝜋 = 𝝅
23
Addition
When adding same signs, we ___________, and
when adding opposite signs, we ____________.
ex:
−25 + −13 = −𝟑𝟖
−25 + 13 = −𝟏𝟐
24
Addition
add
When adding same signs, we ___________,
and
when adding opposite signs, we ____________.
ex:
−25 + −13 = −𝟑𝟖
−25 + 13 = −𝟏𝟐
25
Addition
add
When adding same signs, we ___________,
and
subtract
when adding opposite signs, we ____________.
ex:
−25 + −13 = −𝟑𝟖
−25 + 13 = −𝟏𝟐
26
Addition
add
When adding same signs, we ___________,
and
subtract
when adding opposite signs, we ____________.
Keep same sign
ex:
−25 + −13 = −𝟑𝟖
25 + 13
−25 + 13 = −𝟏𝟐
27
Addition
add
When adding same signs, we ___________,
and
subtract
when adding opposite signs, we ____________.
Keep same sign
ex:
−25 + −13 = −𝟑𝟖
25 + 13
−25 + 13 = −𝟏𝟐
Keep sign of −25
25 − 13
28
Addition Properties
1. 𝑎 + 0 = 𝑎
2. 𝑎 + −𝑎 = 0
Note: 𝑎 and −𝑎 are called ___________________.
ex: The additive inverse of −4 is __________.
ex: The additive inverse of 3 is ___________.
ex: Find −𝑥 if 𝑥 = −6.
−𝑥 = − −6 = 𝟔
29
Addition Properties
1. 𝑎 + 0 = 𝑎
2. 𝑎 + −𝑎 = 0
additive inverses
Note: 𝑎 and −𝑎 are called ___________________.
ex: The additive inverse of −4 is __________.
ex: The additive inverse of 3 is ___________.
ex: Find −𝑥 if 𝑥 = −6.
−𝑥 = − −6 = 𝟔
30
Addition Properties
1. 𝑎 + 0 = 𝑎
2. 𝑎 + −𝑎 = 0
additive inverses
Note: 𝑎 and −𝑎 are called ___________________.
𝟒
ex: The additive inverse of −4 is __________.
ex: The additive inverse of 3 is ___________.
ex: Find −𝑥 if 𝑥 = −6.
−𝑥 = − −6 = 𝟔
31
Addition Properties
1. 𝑎 + 0 = 𝑎
2. 𝑎 + −𝑎 = 0
additive inverses
Note: 𝑎 and −𝑎 are called ___________________.
𝟒
ex: The additive inverse of −4 is __________.
−𝟑
ex: The additive inverse of 3 is ___________.
ex: Find −𝑥 if 𝑥 = −6.
−𝑥 = − −6 = 𝟔
32
Addition Properties
1. 𝑎 + 0 = 𝑎
2. 𝑎 + −𝑎 = 0
additive inverses
Note: 𝑎 and −𝑎 are called ___________________.
𝟒
ex: The additive inverse of −4 is __________.
−𝟑
ex: The additive inverse of 3 is ___________.
ex: Find −𝑥 if 𝑥 = −6.
−𝑥 = − −6 = 𝟔
33
Addition Properties
1. 𝑎 + 0 = 𝑎
2. 𝑎 + −𝑎 = 0
additive inverses
Note: 𝑎 and −𝑎 are called ___________________.
𝟒
ex: The additive inverse of −4 is __________.
−𝟑
ex: The additive inverse of 3 is ___________.
ex: Find −𝑥 if 𝑥 = −6.
−𝑥 = − −6 = 𝟔
34
Addition Properties
1. 𝑎 + 0 = 𝑎
2. 𝑎 + −𝑎 = 0
additive inverses
Note: 𝑎 and −𝑎 are called ___________________.
𝟒
ex: The additive inverse of −4 is __________.
−𝟑
ex: The additive inverse of 3 is ___________.
ex: Find −𝑥 if 𝑥 = −6.
−𝑥 = − −6 = 𝟔
35
Subtracting
We can define subtraction in terms of addition:
𝒂 − 𝒃 = 𝒂 + (−𝒃)
ex:
6 − 13 = 6 + −13 = −𝟕
5.1 − −4.2 = 5.1 + 4.2 = 𝟗. 𝟑
36
Subtracting
We can define subtraction in terms of addition:
𝒂 − 𝒃 = 𝒂 + (−𝒃)
ex:
6 − 13 = 6 + −13 = −𝟕
5.1 − −4.2 = 5.1 + 4.2 = 𝟗. 𝟑
37
Subtracting
We can define subtraction in terms of addition:
𝒂 − 𝒃 = 𝒂 + (−𝒃)
ex:
6 − 13 = 6 + −13 = −𝟕
5.1 − −4.2 = 5.1 + 4.2 = 𝟗. 𝟑
38
Subtracting
We can define subtraction in terms of addition:
𝒂 − 𝒃 = 𝒂 + (−𝒃)
ex:
6 − 13 = 6 + −13 = −𝟕
5.1 − −4.2 = 5.1 + 4.2 = 𝟗. 𝟑
39
Subtracting
We can define subtraction in terms of addition:
𝒂 − 𝒃 = 𝒂 + (−𝒃)
ex:
6 − 13 = 6 + −13 = −𝟕
5.1 − −4.2 = 5.1 + 4.2 = 𝟗. 𝟑
40
Multiplying
When multiplying ___________ signs, the result is
_________________.
When multiplying ___________ signs, the result is
_________________.
ex:
7(−3) =
−4 −2 =
−5 2 =
−52 =
2 4
−
3
=
41
Multiplying
When multiplying ___________
signs, the result is
same
_________________.
When multiplying ___________ signs, the result is
_________________.
ex:
7(−3) =
−4 −2 =
−5 2 =
−52 =
2 4
−
3
=
42
Multiplying
When multiplying ___________
signs, the result is
same
positive
_________________.
When multiplying ___________ signs, the result is
_________________.
ex:
7(−3) =
−4 −2 =
−5 2 =
−52 =
2 4
−
3
=
43
Multiplying
When multiplying ___________
signs, the result is
same
positive
_________________.
opposite signs, the result is
When multiplying ___________
_________________.
ex:
7(−3) =
−4 −2 =
−5 2 =
−52 =
2 4
−
3
=
44
Multiplying
When multiplying ___________
signs, the result is
same
positive
_________________.
opposite signs, the result is
When multiplying ___________
negative
_________________.
ex:
7(−3) =
−4 −2 =
−5 2 =
−52 =
2 4
−
3
=
45
Multiplying
When multiplying ___________
signs, the result is
same
positive
_________________.
opposite signs, the result is
When multiplying ___________
negative
_________________.
ex:
7(−3) = −𝟐𝟏
−4 −2 =
−5 2 =
−52 =
2 4
−
3
=
46
Multiplying
When multiplying ___________
signs, the result is
same
positive
_________________.
opposite signs, the result is
When multiplying ___________
negative
_________________.
ex:
7(−3) = −𝟐𝟏
−4 −2 = 𝟖
−5 2 =
−52 =
2 4
−
3
=
47
Multiplying
When multiplying ___________
signs, the result is
same
positive
_________________.
opposite signs, the result is
When multiplying ___________
negative
_________________.
ex:
7(−3) = −𝟐𝟏
−4 −2 = 𝟖
−5 2 = −5 −5
−52 =
2 4
−
3
=
48
Multiplying
When multiplying ___________
signs, the result is
same
positive
_________________.
opposite signs, the result is
When multiplying ___________
negative
_________________.
ex:
7(−3) = −𝟐𝟏
−4 −2 = 𝟖
−5 2 = −5 −5 = 𝟐𝟓
−52 =
2 4
−
3
=
49
Multiplying
When multiplying ___________
signs, the result is
same
positive
_________________.
opposite signs, the result is
When multiplying ___________
negative
_________________.
ex:
7(−3) = −𝟐𝟏
−4 −2 = 𝟖
−5 2 = −5 −5 = 𝟐𝟓
−52 = −(5 ⋅ 5)
2 4
−
3
=
50
Multiplying
When multiplying ___________
signs, the result is
same
positive
_________________.
opposite signs, the result is
When multiplying ___________
negative
_________________.
ex:
7(−3) = −𝟐𝟏
−4 −2 = 𝟖
−5 2 = −5 −5 = 𝟐𝟓
−52 = − 5 ⋅ 5 = −𝟐𝟓
2 4
−
3
=
51
Multiplying
When multiplying ___________
signs, the result is
same
positive
_________________.
opposite signs, the result is
When multiplying ___________
negative
_________________.
ex:
7(−3) = −𝟐𝟏
−4 −2 = 𝟖
−5 2 = −5 −5 = 𝟐𝟓
−52 = − 5 ⋅ 5 = −𝟐𝟓
2 4
−
3
=
2
−
3
2
−
3
2
−
3
2
−
3
52
Multiplying
When multiplying ___________
signs, the result is
same
positive
_________________.
opposite signs, the result is
When multiplying ___________
negative
_________________.
ex:
7(−3) = −𝟐𝟏
−4 −2 = 𝟖
−5 2 = −5 −5 = 𝟐𝟓
−52 = − 5 ⋅ 5 = −𝟐𝟓
2 4
−
3
=
2
−
3
2
−
3
2
−
3
2
−
3
=
𝟏𝟔
𝟖𝟏
53
Multiplication Properties
1. 𝑎 ⋅ 0 = 0
ex:
2. 𝑎 ⋅ 1 = 𝑎
ex:
54
Multiplication Properties
1. 𝑎 ⋅ 0 = 0
ex: 𝜋 ⋅ 0 = 0
2. 𝑎 ⋅ 1 = 𝑎
ex:
55
Multiplication Properties
1. 𝑎 ⋅ 0 = 0
ex: 𝜋 ⋅ 0 = 0
0⋅
𝑥 2 −3𝑥+2
23
=0
2. 𝑎 ⋅ 1 = 𝑎
ex:
56
Multiplication Properties
1. 𝑎 ⋅ 0 = 0
ex: 𝜋 ⋅ 0 = 0
0⋅
𝑥 2 −3𝑥+2
23
=0
2. 𝑎 ⋅ 1 = 𝑎
ex:
3⋅1= 3
57
Multiplication Properties
1. 𝑎 ⋅ 0 = 0
ex: 𝜋 ⋅ 0 = 0
0⋅
𝑥 2 −3𝑥+2
23
=0
2. 𝑎 ⋅ 1 = 𝑎
ex:
3⋅1= 3
1 ⋅ 𝑥2 = 𝑥2
58
Dividing
We can define division in terms of multiplication:
𝟏
𝒂÷𝒃=𝒂⋅
𝒃
1
and
𝑏
Note: 𝑏
are ________________ of each
other (also called _________________________)
ex:
59
Dividing
We can define division in terms of multiplication:
𝟏
𝒂÷𝒃=𝒂⋅
𝒃
1
and
𝑏
reciprocals
Note: 𝑏
are ________________
of each
other (also called _________________________)
ex:
60
Dividing
We can define division in terms of multiplication:
𝟏
𝒂÷𝒃=𝒂⋅
𝒃
1
and
𝑏
reciprocals
Note: 𝑏
are ________________
of each
multiplicative inverses
other (also called _________________________)
ex:
61
Dividing
We can define division in terms of multiplication:
𝟏
𝒂÷𝒃=𝒂⋅
𝒃
1
and
𝑏
reciprocals
Note: 𝑏
are ________________
of each
multiplicative inverses
other (also called _________________________)
1
2
3
ex:
2
3
3
2
and are reciprocals
62
Dividing
We can define division in terms of multiplication:
𝟏
𝒂÷𝒃=𝒂⋅
𝒃
1
and
𝑏
reciprocals
Note: 𝑏
are ________________
of each
multiplicative inverses
other (also called _________________________)
1
2
3
ex:
2
3
3
2
and are reciprocals
What’s the reciprocal of −4?
63
Dividing
We can define division in terms of multiplication:
𝟏
𝒂÷𝒃=𝒂⋅
𝒃
1
and
𝑏
reciprocals
Note: 𝑏
are ________________
of each
multiplicative inverses
other (also called _________________________)
1
2
3
ex:
2
3
3
2
and are reciprocals
What’s the reciprocal of −4?
𝟏
−
𝟒
64
Dividing
ex:
3
−
4
÷
9
−
5
=
65
Division Properties
1. 0 ÷ 𝑎 = 0
(if 𝑎 ≠ 0)
2. 𝑎 ÷ 0 is __________________
66
Division Properties
1. 0 ÷ 𝑎 = 0
(if 𝑎 ≠ 0)
undefined
2. 𝑎 ÷ 0 is __________________
67
Order of Operations
ex: Simplify.
4 − 72 + 8 ÷ 2 −3
2
=
68
Order of Operations
ex: Simplify.
13−3 −2 4
3− 6−10
=
69
Distributive Property
𝑎 𝑏 + 𝑐 = ________________
ex: Simplify.
−2 3𝑥 + 5 =
𝑥−2 ⋅4=
3 𝑥 − 3 + 2𝑦 =
70
Distributive Property
𝑎 𝑏 + 𝑐 = ________________
𝒂𝒃 + 𝒂𝒄
ex: Simplify.
−2 3𝑥 + 5 =
𝑥−2 ⋅4=
3 𝑥 − 3 + 2𝑦 =
71
Distributive Property
𝑎 𝑏 + 𝑐 = ________________
𝒂𝒃 + 𝒂𝒄
ex: Simplify.
−2 3𝑥 + 5 =
𝑥−2 ⋅4=
3 𝑥 − 3 + 2𝑦 =
72
Distributive Property
𝑎 𝑏 + 𝑐 = ________________
𝒂𝒃 + 𝒂𝒄
ex: Simplify.
−2 3𝑥 + 5 = −2 ⋅ 3𝑥 + −2 ⋅ 5
𝑥−2 ⋅4=
3 𝑥 − 3 + 2𝑦 =
73
Distributive Property
𝑎 𝑏 + 𝑐 = ________________
𝒂𝒃 + 𝒂𝒄
ex: Simplify.
−2 3𝑥 + 5 = −2 ⋅ 3𝑥 + −2 ⋅ 5
= −6𝑥 + (−10)
𝑥−2 ⋅4=
3 𝑥 − 3 + 2𝑦 =
74
Distributive Property
𝑎 𝑏 + 𝑐 = ________________
𝒂𝒃 + 𝒂𝒄
ex: Simplify.
−2 3𝑥 + 5 = −2 ⋅ 3𝑥 + −2 ⋅ 5
= −6𝑥 + (−10)
= −𝟔𝒙 − 𝟏𝟎
𝑥−2 ⋅4=
3 𝑥 − 3 + 2𝑦 =
75
Distributive Property
𝑎 𝑏 + 𝑐 = ________________
𝒂𝒃 + 𝒂𝒄
ex: Simplify.
−2 3𝑥 + 5 = −2 ⋅ 3𝑥 + −2 ⋅ 5
= −6𝑥 + (−10)
= −𝟔𝒙 − 𝟏𝟎
𝑥−2 ⋅4=
3 𝑥 − 3 + 2𝑦 =
76
Distributive Property
𝑎 𝑏 + 𝑐 = ________________
𝒂𝒃 + 𝒂𝒄
ex: Simplify.
−2 3𝑥 + 5 = −2 ⋅ 3𝑥 + −2 ⋅ 5
= −6𝑥 + (−10)
= −𝟔𝒙 − 𝟏𝟎
𝑥−2 ⋅4=𝑥⋅4−2⋅4
3 𝑥 − 3 + 2𝑦 =
77
Distributive Property
𝑎 𝑏 + 𝑐 = ________________
𝒂𝒃 + 𝒂𝒄
ex: Simplify.
−2 3𝑥 + 5 = −2 ⋅ 3𝑥 + −2 ⋅ 5
= −6𝑥 + (−10)
= −𝟔𝒙 − 𝟏𝟎
𝑥−2 ⋅4=𝑥⋅4−2⋅4
= 𝟒𝒙 − 𝟖
3 𝑥 − 3 + 2𝑦 =
78
Distributive Property
𝑎 𝑏 + 𝑐 = ________________
𝒂𝒃 + 𝒂𝒄
ex: Simplify.
−2 3𝑥 + 5 = −2 ⋅ 3𝑥 + −2 ⋅ 5
= −6𝑥 + (−10)
= −𝟔𝒙 − 𝟏𝟎
𝑥−2 ⋅4=𝑥⋅4−2⋅4
= 𝟒𝒙 − 𝟖
3 𝑥 − 3 + 2𝑦 =
79
Distributive Property
𝑎 𝑏 + 𝑐 = ________________
𝒂𝒃 + 𝒂𝒄
ex: Simplify.
−2 3𝑥 + 5 = −2 ⋅ 3𝑥 + −2 ⋅ 5
= −6𝑥 + (−10)
= −𝟔𝒙 − 𝟏𝟎
𝑥−2 ⋅4=𝑥⋅4−2⋅4
= 𝟒𝒙 − 𝟖
3 𝑥 − 3 + 2𝑦 = 𝟑𝒙 − 𝟗 + 𝟔𝒚
80
Combining Like Terms
The parts of an algebraic expression separated
by addition are called _____________.
ex: 7𝑥 − 9𝑦 + 𝑧 − 3
81
Combining Like Terms
The parts of an algebraic expression separated
terms
by addition are called _____________.
ex: 7𝑥 − 9𝑦 + 𝑧 − 3
82
Combining Like Terms
The parts of an algebraic expression separated
terms
by addition are called _____________.
ex: 7𝑥 − 9𝑦 + 𝑧 − 3
7𝑥 + −9𝑦 + 𝑧 + (−3)
83
Combining Like Terms
The parts of an algebraic expression separated
terms
by addition are called _____________.
ex: 7𝑥 − 9𝑦 + 𝑧 − 3
7𝑥 + −9𝑦 + 𝑧 + (−3)
4 terms: 𝟕𝒙, −𝟗𝒚, 𝒛, and −𝟑
84
Combining Like Terms
The numerical part of a term is called the
_____________________.
ex:
85
Combining Like Terms
The numerical part of a term is called the
coefficient
_____________________.
ex:
86
Combining Like Terms
The numerical part of a term is called the
coefficient
_____________________.
ex:
Term
−9𝑦
𝑧
−3
Coefficient
87
Combining Like Terms
The numerical part of a term is called the
coefficient
_____________________.
ex:
Term
−9𝑦
𝑧
−3
Coefficient
−𝟗
88
Combining Like Terms
The numerical part of a term is called the
coefficient
_____________________.
ex:
Term
−9𝑦
𝑧
−3
Coefficient
−𝟗
𝟏
89
Combining Like Terms
The numerical part of a term is called the
coefficient
_____________________.
ex:
Term
−9𝑦
𝑧
−3
Coefficient
−𝟗
𝟏
−𝟑
90
Combining Like Terms
The parts of a term that are multiplied are called
______________________.
91
Combining Like Terms
The parts of a term that are multiplied are called
factors
______________________.
ex: The factors of −9𝑦 are: _______________
92
Combining Like Terms
The parts of a term that are multiplied are called
factors
______________________.
−𝟗 and 𝒚
ex: The factors of −9𝑦 are: _______________
93
Combining Like Terms
The parts of a term that are multiplied are called
factors
______________________.
−𝟗 and 𝒚
ex: The factors of −9𝑦 are: _______________
ex: The factors of 3 𝑥 + 1 𝑦 are:
___________________________
94
Combining Like Terms
The parts of a term that are multiplied are called
factors
______________________.
−𝟗 and 𝒚
ex: The factors of −9𝑦 are: _______________
ex: The factors of 3 𝑥 + 1 𝑦 are:
𝟑, 𝒙 + 𝟏, and 𝒚
___________________________
95
Combining Like Terms
Terms with the same variable factors are called
______________________.
96
Combining Like Terms
Terms with the same variable factors are called
like terms
______________________.
97
Combining Like Terms
Terms with the same variable factors are called
like terms
______________________.
ex:
3𝑥 and 7𝑥 are like terms.
98
Combining Like Terms
Terms with the same variable factors are called
like terms
______________________.
ex:
3𝑥 and 7𝑥 are like terms.
7𝑦 2 and 𝑦 2 are like terms.
99
Combining Like Terms
Terms with the same variable factors are called
like terms
______________________.
ex:
3𝑥 and 7𝑥 are like terms.
7𝑦 2 and 𝑦 2 are like terms.
2𝑥 and 4𝑥 2 are not like terms.
100
Combining Like Terms
We can combine like terms to simplify.
ex: 3𝑥 + 7𝑥 =
7𝑦 2 − 𝑦 2 =
101
Combining Like Terms
We can combine like terms to simplify.
ex: 3𝑥 + 7𝑥 = 3 + 7 𝑥
7𝑦 2 − 𝑦 2 =
102
Combining Like Terms
We can combine like terms to simplify.
ex: 3𝑥 + 7𝑥 = 3 + 7 𝑥
distributive property (in “reverse”)
7𝑦 2 − 𝑦 2 =
103
Combining Like Terms
We can combine like terms to simplify.
ex: 3𝑥 + 7𝑥 = 3 + 7 𝑥 = 𝟏𝟎𝒙
distributive property (in “reverse”)
7𝑦 2 − 𝑦 2 = 6𝑦 2
104
Combining Like Terms
We can combine like terms to simplify.
ex: 3𝑥 + 7𝑥 = 3 + 7 𝑥 = 𝟏𝟎𝒙
distributive property (in “reverse”)
7𝑦 2 − 𝑦 2 = 𝟔𝒚𝟐
7−1
105
Combining Like Terms
ex: Simplify.
4 7𝑥 − 3 − 10𝑥
− 3𝑥 2 − 7𝑥 − 4
8𝑥 + 2 5 − 𝑥 − 3
106
Practice Problems
- Work individually…
- …but feel free to get help from neighbors.
- Raise hand to get help from me.
- Remember: the harder you work here, the easier
your homework will be.
107
Practice Problems
- Work individually…
- …but feel free to get help from neighbors.
- Raise hand to get help from me.
- Remember: the harder you work here, the easier
your homework will be.
108
Practice Problems
- Work individually…
- …but feel free to get help from neighbors.
- Raise hand to get help from me.
- Remember: the harder you work here, the easier
your homework will be.
109
Practice Problems
- Work individually…
- …but feel free to get help from neighbors.
- Raise hand to get help from me.
- Remember: the harder you work here, the easier
your homework will be.
110
Practice Problems
- Work individually…
- …but feel free to get help from neighbors.
- Raise hand to get help from me.
- Remember: the harder you work here, the easier
your homework will be.
111