Download Mathematics 90 (3634) ____Solutions__________ CA (Class

Mathematics 90 (3634)
CA (Class Addendum) 7: Property Identification
Mt. San Jacinto College
Menifee Valley Campus
Spring 2009
____Solutions__________
Name
This class addendum is worth a maximum of five (5) points. It is due no later than the end of
class on Friday, 8 May.
Attached you will find a linear equation solved in great detail. For each equation, state the
definition or property from the list below that makes the equation equivalent to its predecessor.
Definition of Subtraction If a and b are real numbers, then
a - b = a + (-b)
The Commutative Property of Addition If a and b are real numbers, then
a+b = b+a
The Commutative Property of Multiplication If a and b are real numbers, then
a b  b a
The Associative Property of Addition If a, b and c are real numbers, then
(a + b) + c = a + (b + c)
The Associative Property of Multiplication If a, b and c are real number, then
( a  b)  c  a  (b  c)
The Addition Property of Zero If a is a real number, then
a+0 =a
or
0+a = a
The Multiplication Property of Zero If a is a real number, then
a0  0
or
0a  0
The Multiplication Property of One If a is a real number, then
a 1  a
or
1 a
 a
The Inverse Property of Addition If a is a real number, then
a + (-a) = 0 or
-a + a = 0
The Inverse Property of Multiplication If a is a real number and a is not zero, then
a
1
a
 1
or
1
a  1 , a  0
a
The Distributive Property If a, b and c are real numbers, then
a  ( b  c )  a b  a  c
or
(b  c )a  ba  c a
The Addition Property of Equations The same number can be added to each side of an equation
without changing the solution to the equation. In symbols, if a = b and c is any number, then
a+c = b+c
The Multiplication Property of Equations Each side of an equation can be multiplied by the
same nonzero number without changing the solution to the equation. In symbols, if a = b and c
is any nonzero number, then
ac  bc , c  0
The Closure Property of Addition If a and b are real numbers, then so is the number a + b.
The Closure Property of Multiplication If a and b are real numbers, then so is the number
a  b.
Grading
There is a maximum of 5 points possible for this assignment. Each incorrect answer will
subtract one-half (1/2) point from your total. Therefore, if you miss 10 (or more) questions, you
will receive zero points.
NOTE: When more than one correct answer exists, the alternate answer is contained in
parentheses. You need only provide one correct answer per line for full credit.
Solve.
3[2 – 4(y – 1)] = 3(2y + 8)
____________Given______________
3[2 – 4(y + [-1])] = 3(2y + 8)
1.Definition of Subtraction
3[2 + (-4)(y + [-1])] = 3(2y + 8)
2.Definition of Subtraction
3[2 + (-4)(y) + (-4)(-1)] = 3(2y + 8)
3.Distributive Property
3[2 + (-4y) + (-4)(-1)] = 3(2y + 8)
4.Closure of Multiplication
3[2 + [-4y] + 4] = 3(2y + 8)
5.Closure of Multiplication
3[2 + ([-4y] + 4)] = 3(2y + 8)
6.Associative Prop. of Addition
3[2 + (4 + [-4y])] = 3(2y + 8)
7.Commutative Prop. of Addition
3[2 + 4 + [-4y]] = 3(2y + 8)
8. Associative Prop. of Addition
3[6 + (-4y)] = 3(2y + 8)
9.Closure of Addition
3[6] + 3[-4y] = 3(2y + 8)
10.Distributive Property
18 + 3[-4y] = 3(2y + 8)
11.Closure of Multiplication
18 + [3(-4)]y = 3(2y + 8)
12.Associative Prop. of Mult.
18 + (-12y) = 3(2y + 8)
13.Closure of Multiplication
18 + (-12y) = 3(2y) + 3(8)
14.Distributive Property
18  (12 y )  (3  2) y  3(8)
15.Associative Prop. of Mult.
18 + (-12y) = 6y + 3(8)
16.Closure of Multiplication
18 + (-12y) = 6y + 24
17. Closure of Multiplication
18 + (-12y) + 12y = 6y + 24 + 12y
18.Addition Prop. of Equations
18 + [(-12y) + 12y] = 6y + 24 + 12y
19.Associative Prop. of Addition
18 + [(-12 + 12)y] = 6y + 24 + 12y
20.Distributive Property
18 + [0y] = 6y + 24 + 12y
21.Inverse Property of Addition
(Closure of Addition)
18 + 0 = 6y + 24 + 12y
22.Multiplication Prop. of Zero
(Closure of Multiplication)
18 = 6y + 24 + 12y
23.Addition Property of Zero
(Closure of Addition)
18 = 6y + [24 + 12y]
24.Associative Prop. of Addition
18 = 6y + [12y + 24]
25.Commutative Prop. of Addition
18 = 6y + 12y + 24
26.Associative Prop. of Addition
18 = (6 + 12)y + 24
27.Distributive Property
18 = 18y + 24
28.Closure of Addition
18 + (-24) = 18y + 24 + (-24)
29.Addition Prop. of Equations
-6 = 18y + 24 + (-24)
30.Closure of Addition
-6 = 18y + [24 + (-24)]
31.Associative Prop. of Addition
-6 = 18y + 0
32.Inverse Property of Addition
(Closure of Addition)
-6 = 18y
33.Addition Property of Zero
(Closure of Addition)
 6
1
1
 18 y 
18
18
34. Multiplication Property of Eqns.

1
1
 18 y 
3
18
35.Closure of Multiplication

1
 1
 18  y  
3
 18 
36.Associative Prop. of Mult.

1
 1 
 18   y 
3
 18 
37.Commutative Prop. of Mult.

1
1
 18   y
3
18
38.Associative Prop. of Mult.

1
 1 y
3
39. Inverse Property of Mult.
(Closure of Multiplication)

1
y
3
40. Multiplication Property of One
(Closure of Multiplication)