____
7 ME
J
Lesson 5 Reteach
DATE
PERIOD
Algebra: Properties
Property
Symbols
Commutative
a+b=b+cz
a.b=b’a
Associative
(a+6+c=a+(b+c)
(a b)•c=a’b’c)
Identity
ct+0—i
a’l=a
.‘
Numbers
5+3=3+5
5.3=3.5
(2+3)+4=2+(3±4)
(23).4=2.(3.4)
5+0=5
5.1=5
Example 1
Determine whether 6 + (4 + 3) and (6 + 4) + 3 arc equivalent.
The numbers are grouped differenth They are equivalent by the Associative Property. So.
6 + (4 + 3)
=
(6 + 4) + 3.
Use the properties to make mental math easier.
Example 2
The formula for the perimeter of a triangle is P a + b + c, where a,
b, and c are side lengths. Find the perimeter of a triangle where
a= 12,b5,andc=8.
F
F
P
P
=
=
=
=
a + b + c
1.2 + 5 + S
Write the formula.
Replace a with 12, b with 5, and c with 8.
12 ± 8 + 5
25 units
Commutative Property
Exerdses
Determine whether the two expressions are equivalent. If so, tell
what property is applied. If not, explain why
1. 9. 1 and 9 yes; Identity
Property
3. 6
(3
2) and (6
—
3)
2. 7. 3 and 3
.
7 yes;
Commutative
Property
-
2
no; the first expression equals
5 and the second equals 1.
4. (10. 2). 5 and 10
.
(2
.
5)
yes; Associative Property
5. The formula for the volume of a rectangular prism is V = 1w/i. where [is the length. iv
is the width, and h is the height. Find the volume of a rectangular prism, in cubic units.
if the length is 8 units, the width is 11 units, and the height is 10 units. Use the
Commutative Property
V=15.8.2=1528=3Q8=24Ocubicunits
Course I
Chapter 6 Expressions
97
_____
NAME
DATE
PERIOD
Lesson 5 Skills Practice
Algebra: Properties
Determine whether the two expressions are equivalent. If so, tell
what property is applied. If not, explain why.
1. 2.(3’7)and(2.3).7
2. 6±3and3+6
yes; Associative Property
3. 26
—
i9
7) and (26
—
—
9)
7
no; the expressions equal
24 and 10.
5. 7
.
2 and 2
.
7
yes; Commutative Property
yes; Commutative Property
4. 18. 1 and 18
yes; Identity Property
6. 6
—
(4
—
1) and (6
—
4)
1
no; the expressions equal 3 and 1.
(3
t
(-
..:
7. 7 + 0 and 7
—
,
(.
8. 0 + 12 arid 0
yes; Identity Property
9. 625 + 281 and 281 + 625
yes; Commutative Property
11. 2 ± (8 + 2) and (2 + 8) + 2
yes; Associative Property
no; the expressions equal
12 and 0.
10. (12. 18) .5 and 12. (18.5)
yes; Associative Property
12. 40 ÷ 10 and 10 ÷ 40
no; the expressions equal 4 and
cL
31
‘5
-)O1
4-.
Qrn3dC.±)
Use one or more properties to rewrite each expression as an
expression that does not use parentheses.
6
.
3
l
(pl)• p.6
14.(a+5)+23
a+28
.\1/
15.7.(v.3)y.2l
16.(b±4)+17 b+21
F C73
(
17. 6 ± x + 50)
C
98
2
E%
t 1
uL
x +56
18. (v200). 2 y.400
y
((3()
Course I
Chapter 6 Expressions
______
NAME
DATE
PERIOD
Lesson 6 Reteach
The Distributive Property
• To multiply a sum by a number, multiply each addend by the number outside the parentheses.
• a(b ± c) = ab
ac
• (b + c)a
ba
=
±
ca
Example 1
Find 6 x 38 mentally using the Distributive Propert
6 x 38 = 6(30 + 8)
Write38as3O+8.
= 6(30) + 6(8)
Distributive Property
= 180 + 48
Multiply mentally.
= 228
Add,
So. 6 x 38
228.
=
Example 2
Use the Distributive Property to rewrite 4(x + 3).
4(x ± 3) = 4(x) + 4(3)
Distributive Property
= 4x + 12
Multiply.
So, 4(x + 3) can be rewritten as 4x + 12.
Exercises
Find each product mentally. Show the steps you used.
L4x82
4 x 82
2.9x26
4(80 + 2)
= 4(80) + 4(2)
=320+8
= 328
9 x 26
=
3.12x44
9(20 + 6)
= 9(20) + 9(6)
=180+54
=
4.8x5.7
12 x 44
12(40 + 4)
12(40) + 12(4)
=480+48
=528
8 x 5.7
=
=
8(5 + 0.7)
= 8(5) + 8(0.7)
=40+5.6
=45.6
=
Use the Distributive Property to rewrite each algebraic expression.
5. 5(y
+ 4)
5y + 20
6. (7 ±
rh,)
1)3
73
8. (b + 2)9 9b + 18
21 + 3r
7. 12(x
7
9. 4(4 ± a) 16 +
4a
1O 9(7 ± a) 63 + 9v
‘‘
=.
•‘
I
I
J74.
12x +60
5)
ii(}
/
11
±
I
.,
,_)/
/
99
-
—‘I
/
I
‘
(I
‘-
—
7/
I
‘7
___________
_
_
NAME
_____________
DATE
_______
PERIOD
Lesson 6 Skills Practice
The Distributive Property
Find each product mentall Show the steps you used.
L3x78
t7x74
3(70)-i-3(8)=234
7(70)+7(4)=518
‘19c>-taa
aai
3.8x92
4.6x57
8(90)+8(2)=736
6(50)+6(7)=342
.3cc tQ.
73(t
5.15x4
6.12x51
15(2)+15(f)=40
7.6x5.2
6(5) + 6(0.2)
12(5)-i- 12(4)=62
8. 4x9.4
=
31.2
4(9) + 4(0.4)
=
3Z6
3Otjc
37J
Use the Distributive Property to rewrite each algebraic expression.
L7(y+2)7y-l-14
1O.(8+r)432-l-4r
11.8Cz+9)8x+72
9(SfYrWc)
3ai-qr
12.(b+5)12 12b-I-60
18.4(2+a)8-l-4a
0/lID
14. 7(6+u)
J
42+7v
JWb)ti1o)
15. (b
—
5)15 15b + 75
3(’5)-3()
15-Sv
15b-75
‘t
100
16. 3(5—u) 15 + 3v
)7x#/&
3(9Xt’1)
j
j
17. 6(11—s) 66-- 6s
CO)-&(&
(60%_3
%
ki9u-aL)
Casual
•
j5(i
Chapter 6 Expressions
‘-3A)
________
NAME
DATE
PERIOD
Lesson 7 Reteach
Equivalent Expressions
• Commutative Property: The order which numbers are added or multiplied does not change the sum
or the product.
•a ± b
=
b ± a or a
.
b
b
.
a.
• Associative Property: The way in which numbers are grouped does not change the sum or the
product.
•(a±b)+c=a ±(b±c)or(a.b).c=a.(b.c)
• Like terms contain the same variables. Examples: 2y, y, and 7y are all like terms. but 4x is not.
Example 1
Simplify the expression 16 + (v + 4).
16 + (v + 4)
16 + (4 ± ct
= (16
4) ± v
= 20 + v
Commutative Property
Associative Property
Add.
So, 16 + (u + 4) in simplified form is 20 + v.
Example 2
Simplify the expression 3x + (6y + 2x).
3x+(6y+2x)=3x+(2x+6y)
= (3x + 2x) + 6y
= 5x ± 6’
E
Commutative Property
Associative Property
Combine like terms.
So, 3x + (6y + 2s) in simplified form is 5x + 6y.
Exercises
Simplify each expression. JusLify each step.
2. 6
1.5± x + 3
io +
(8+x
K
4. 8x
3. (b + 10) + 15
± (x + 4)
Cf2A P
5. i12
± 5 ± 2x
tX
/
± 2u)
L /2/2
3
lOx + 5
5+2u
L
7. 9x + (4z + 3x)
(12x + 4z
b + 25
‘
ii
I
18p+8
iv
Q_c’I
:/‘
8. (8z + .12x) + (2z + 7x)
(lOz + 19x
9. Sy +
4z + 7z
5y + liz
2
/
Course I
Chapter 6 Expressions
I5) ,/
6. lip ± S + 7p
S
0
(
NAME
DATE
PERIOD
Lesson 7 Skills Practice
Equivalent Expressions
Simplify each expression.
1.x+4±3x
2. 3±.x+6
; 4x + 4
3. 15 + (6 + x)
+ x
21 + x
7
.o t)
k±
4. (6 + x) + 9
5. x + 2 + Sx
5±x
\
yr .k
9x + 2
I
/
8. 15
29y+5x
5
L.
6. (4x ± 3v) + 23r
[4x + 26y
x -(3y
:c)
7. (25v ± 5x) ± 4)’
.
(5
.
x)
28x
1k.
11. x ± 2 + x
12x + 16
12. 5
2x + 2
•
3
.
x. 10
50x
(&icx
/
x)
oLy)
9. 7(4x)
S75x
10. 8x + (16 + 4x)
13. ç17
)c
4-
14. & + 17’ + 9x
15.3x + (24x + 8)
7
K51x
‘9 17x + 17y
16. 4(15x)
17. 2x + 8 + x
60x
27x + 8
•/:)f:.
18. (5x ± 9v) + 32x
3x±8
©
(37x+9y
iX:t
;4j).
)Lf
n
Y
A car company charges x dollars to znt a car
plus any extra options shown in the table. Use
the information to answer Exercises 19 and 20.
19. Three people each rented a car with insurance
and one more person rented a car with a car
wash. Write an expression that represents the
total cost of the car rentals and extra options.
[
Extra Options
Cost
Car Seat
$50
Insurance
$75
Car Wash
S15
3(75) + x + 15 + x + x
20. Simplify your expression. Justify each step.
240 + 3x
Course I
Chapter 6 Expressions