Download 1 Mathematics 90 (3634) ___Solutions_______ CA 6: Property

Mathematics 90 (3634)
CA 6: 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, 24 April.
NOTE: You may need to study this entire addendum carefully several times before you
begin the exercises it contains. You may need to study example exercises carefully
several times before you attempt the exercise sets that follow them. Also, the order in
which the exercises occur may not necessarily be the order in which you complete them.
If you find that the solution to a particular exercise eludes you, skip to another one. You
are being given two weeks to complete this handout because you’ll probably need to
study it, attempt some of the exercises and then take a break, continuing with it a day or
two later.
Every exercise included in this and the final class addendum will have the following
instructions:
“For each equation, state the definition or property that makes the
equation equivalent to its predecessor.”
You will be provided with a list of the real number properties. You will be provided with
an equation solved in great detail. Each step in this solution will correspond to an
application of a real number property. Your job is to identify which property was applied
at each step.
You should familiarize yourself with the list of real number properties, the equation and
the steps utilized to solve this equation, and then study the following examples.
Example 1. For equation 2, state the definition or property that makes the equation
equivalent to its predecessor (equation 1).
3 y  (7 )  5  2 y  3  4
1.______________________
3 y  (7 )  5  2 y  [ 3 ]  4
2.______________________
Solution: Our job is to identify which definition or property makes the top equation
equivalent to the bottom one. In other words, assuming these two equations are
equivalent (i.e. they have the same solution set), which property or definition justifies the
difference in the symbols utilized to express each equation?
1
We must first identify exactly where different symbols occur. Notice that the left-hand
sides of the two equations are identical. Nothing changes from equation 1 to equation 2
as we compare the left-hand sides. Therefore, we can ignore the left-hand sides:
3 y  (7 )  5  2 y  3  4
3 y  (7 )  5  2 y  [ 3 ]  4
The change must have occurred on the right-hand side. That is, the real number property
utilized must have changed something about the right-hand side of equation 1 to create
the right-hand side of equation 2. Compare the right-hand sides:
5  2 y  3  4
5  2 y  [  3 ]  4
Notice that as we read both expressions from left to right (like we read text in a book), we
see a five multiplied to a quantity in parentheses. The first term in each set of
parentheses is 2y. In other words, so far, there is no change. However, we see a minus in
the top expression and a plus in the bottom expression. A change has occurred! A three
follows the minus in the top expression. However, a negative three follows the plus in
the bottom expression. This difference in symbols constitutes another change.
Continuing this comparison by reading left to write, we see both expressions are
identical. Therefore, to identify the property utilized to create these symbol changes, we
can probably forget about multiplication by five and addition by 4 since these happen in
both expressions.
Since a minus in the top expression aligns vertically with a plus in the bottom expression,
we’ll need to find a property whose formula contains both of these symbols. For an
appropriate comparison between symbols we find in our expressions and symbols used
for the property formula, we’ll need to focus on symbols surrounding the plus and minus
signs. We’ll start by ignoring multiplication by five, the parentheses, and addition by
four:
5  2 y  3  4
5  2 y  [  3 ]  4
This leaves us with the expression 2y – 3 from the original upper equation and the
expression 2y + [-3] from the original lower equation.
These expressions must be equal since the two original equations are equivalent (more on
this in the NOTE below). Therefore we have:
2y – 3 = 2y + [-3].
2
Which property from the list is expressed by this equation? Letting 2y in the equation
correspond to the letter a and the constant 3 correspond to the letter b, we see that the
equation above takes the form
a – b = a + (-b).
But this is the formula for the Definition of Subtraction!
Therefore, the answer (to line 2) is: The Definition of Subtraction.
NOTE: You have used substitution to solve a 2 by 2 system of linear equations. The
same principle can be used to justify the truth of the equation 2y – 3 = 2y + [-3] above.
Recall that the left-hand sides of equations 1 and 2 are identical. Therefore, we can
substitute the right-hand side of either equation for the left-hand side of the other. This
yields an equation equivalent to
5  2 y  [ 3 ]  4  5  2 y  3  4 .
If we subtract four from both sides and then divide both sides by 5, we have an equation
equivalent to 2y – 3 = 2y + [-3].
Example 2. For equation 8, state the definition or property that makes the equation
equivalent to its predecessor (equation 7).
3 y  (7 )  10 y  [ (15 )  4 ]
7.______________________
3 y  (7 )  10 y  (11 )
8.______________________
Solution: Our job is to identify which definition or property makes the top equation
equivalent to the bottom one. In other words, assuming these two equations are
equivalent (i.e. they have the same solution set), which property or definition justifies the
difference in the symbols utilized to express each equation?
We must first identify exactly where different symbols occur. Notice that the left-hand
sides of the two equations are identical. Nothing changes from equation 1 to equation 2
as we compare the left-hand sides. Therefore, we can ignore the left-hand sides:
3 y  (7 )  10 y  [ (15 )  4 ]
3 y  (7 )  10 y  (11 )
The change must have occurred on the right-hand side. That is, the real number property
utilized must have changed something about the right-hand side of equation 7 to create
the right-hand side of equation 8. Compare the right-hand sides:
3
10 y  [ (15 )  4 ]
10 y  (11)
Notice that as we read both expressions from left to right (like we read text in a book), we
see the term 10y and then a plus sign. In other words, so far, there is no change.
However, in the top expression we next see -15 added to 4. In the bottom expression, we
see the simplification of this sum, -11. A change has occurred!
Ignoring the 10y and plus sign in each expression, we equate the remaining symbols from
equation 7 with those from equation 8:
-15 + 4 = -11.
That’s certainly true! However, our job is to determine which property justifies the truth
of this equation. Consider the following property:
The Closure Property of Addition If a and b are real numbers, then so is the number a + b.
If we let a in this property correspond with -15 and b correspond with 4, the equation -15
+ 4 = -11 and closure property “formula” could be aligned vertically as follows:
-15 + 4 = -11
a +b=a+b
That is, on the left side of the equation -15 + 4 = -11, we see the addends -15
(corresponding to a) and 4 (corresponding to b). These are separate numbers that have
yet to be added. On the right-hand side, we have their sum -11 (corresponding to the
number a + b, the (single number) answer you get when you add the number a to the
number b).
The answer (to line 8) is: The Closure Property of Addition.
Example 3. For equation 9, state the definition or property that makes the equation
equivalent to its predecessor (equation 8).
3 y  (7 )  10 y  (11 )
8.______________________
3 y  (7 )  (3 y )  10 y  (11 )  (3 y )
9.______________________
Solution: Our job is to identify which definition or property makes the top equation
equivalent to the bottom one. In other words, assuming these two equations are
equivalent (i.e. they have the same solution set), which property or definition justifies the
difference in the symbols utilized to express each equation?
We must first identify exactly where different symbols occur. Notice that neither the lefthand sides nor the right-hand sides of the two equations are identical. Both sides of
equation 9 have more writing than these sides exhibited previously (in equation 8). On
4
their left-hand sides, both equations contain the terms 3y and -7. While these are the only
terms on the left-hand side of equation 8, the left-hand side of equation 9 contains the
additional term -3y. On their right-hand sides, both equations contain the terms 10y and 11. However, once again, the right-hand side of equation 9 contains a term not present on
the right-hand side of equation 8: -3y. That is, equation 9 looks like equation 8 with -3y
added to both sides.
Consider the following property:
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
If we let the letter a correspond to the expression 3y + (-7) in equation 9, the letter b
correspond to the expression 10y + (-11) in equation 9 and the letter c correspond to the
term -3y in equation 9, we have the following correspondence between equation 9 and
the formula for the Addition Property of Equations:
3 y  (7 )  (3 y )  10 y  (11 )  (3 y )
a
+ c =
b
+ c
That is, the Addition Property of Equations was utilized to transform equation 8 into
equation 9.
The answer (to line 9) is: The Addition Property of Equations.
You are now ready to begin the exercises. The list of real number properties and
definitions is now provided, followed by the equation and solution steps to be justified.
Real Number Properties and Definitions
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
5
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
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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.
The Equation and Solution (Here’s where you fill in the blanks)
Exercise For each equation, state the definition or property from the list above that
makes the equation equivalent to its predecessor.
There is a maximum of 5 points possible. 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 is possible, the alternate answer is contained
in parentheses. You need only give one of these answers for full credit.
Solve.
3( y  7)  6  5 y  4
_________Given__________
3( y  [7])  6  5 y  4
1.Definition of Subtraction
3( y  [7])  6  5 y  [ 4]
2.Definition of Subtraction
3 y  3[7]  6  5 y  [ 4]
3.Distributive Property
3 y  [21]  6  5 y  [ 4]
4.Closure of Multiplication
3 y  ( [21]  6)  5 y  [ 4]
5.Associative Prop. of Mult.
3 y  (15)  5 y  [ 4]
6.Closure of Addition
3 y  (15)  (3 y )  5 y  [ 4]  (3 y )
7.Add. Property of Equations
3 y  [(15)  (3 y )]  5 y  [ 4]  (3 y )
8.Associative Prop. of Add.
3 y  [(3 y )  (15)]  5 y  [ 4]  (3 y )
9.Commutative Prop. of Add.
3 y  (3 y )  (15)  5 y  [ 4]  (3 y )
10.Associative Prop. of Add.
(3  [3]) y  (15)  5 y  [ 4]  (3 y )
11.Distributive Property
7
0  y  (15)  5 y  [ 4]  (3 y )
12.Inverse Property of Add.
(Closure of Addition)
0  (15)  5 y  [ 4]  (3 y )
13.Multiplication Prop. of Zero
(Closure of Multiplication)
 15  5 y  [ 4]  (3 y )
14.Addition Prop. of Zero
(Closure of Addition)
 15  5 y  ([ 4]  (3 y ))
15.Associative Prop. of Add.
 15  5 y  ([ 3 y ]  (4))
16.Commutative Prop. of Add.
 15  5 y  [ 3 y ]  (4)
17.Associative Prop. of Add.
 15  (5  [ 3]) y  (4)
18.Distributive Property
 15  2 y  (4)
19.Closure of Addition
 15  4  2 y  (4)  4
20.Addition Prop. of Equations
 15  4  2 y  [(4)  4]
21.Associative Prop. of Add.
 15  4  2 y  0
22.Inverse Prop. of Addition
(Closure of Addition)
 15  4  2 y
23.Addition Prop. of Zero
(Closure of Addition)
 11  2 y
24.Closure of Addition
(11) 
1
1
 ( 2y ) 
2
2
25.Mult. Prop. of Equations
 11
1
 ( 2y ) 
2
2
26.Closure of Multiplication
 11
1
 ( y2 ) 
2
2
27.Commutative Prop. of Mult.
 11
 1
 y  2  
2
 2
28.Associative Prop. of Mult.
8
 11
 y 1
2
29.Inverse Prop. of Mult.
(Closure of Multiplication)
 11
 y
2
30.Multiplication Prop. of One
(Closure of Multiplication)
9