Mathematics 96 (3581) CA 6: Property Identification Mt. San Jacinto College Menifee Valley Campus Spring 2013

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1 Mathematics 96 (358) CA 6: Property Identification Mt. San Jacinto College Menifee Valley Campus Spring 203 Name This class addendum is worth a maximum of five (5) points. It is due no later than the end of class on Friday, 3 May. 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. For equation 2, state the definition or property that makes the equation equivalent to its predecessor (equation ). 2y 3 4 ( 7) 5. 2y [ 3] 4 ( 7) 5 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?

2 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 to equation 2 as we compare the left-hand sides. Therefore, we can ignore the left-hand sides: 2y 3 4 ( 7) 5 2y [ 3] 4 ( 7) 5 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 to create the right-hand side of equation 2. Compare the right-hand sides: 2y y [ 3] 4 5 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: 2y y [ 3] 4 5 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

3 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 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 2 [ 3] 4 5 2y y. 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). ( 7) 0y [( 5) 4] 7. ( 7) 0y ( ) 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 to equation 2 as we compare the left-hand sides. Therefore, we can ignore the left-hand sides: ( 7) 0y [( 5) 4] ( 7) 0y ( ) 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

4 0y [( 5) 4] 0y ( ) Notice that as we read both expressions from left to right (like we read text in a book), we see the term 0y and then a plus sign. In other words, so far, there is no change. However, in the top expression we next see -5 added to 4. In the bottom expression, we see the simplification of this sum, -. A change has occurred! Ignoring the 0y and plus sign in each expression, we equate the remaining symbols from equation 7 with those from equation 8: = -. 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 -5 and b correspond with 4, the equation = - and closure property formula could be aligned vertically as follows: = - a + b = a + b That is, on the left side of the equation = -, we see the addends -5 (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 - (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). ( 7) 0y ( ) 8. ( 7) ( ) 0y ( ) ( ) 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

5 their left-hand sides, both equations contain the terms 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 -. On their right-hand sides, both equations contain the terms 0y and -. However, once again, the right-hand side of equation 9 contains a term not present on the right-hand side of equation 8: -. That is, equation 9 looks like equation 8 with - 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 + (-7) in equation 9, the letter b correspond to the expression 0y + (-) in equation 9 and the letter c correspond to the term - in equation 9, we have the following correspondence between equation 9 and the formula for the Addition Property of Equations: ( 7) ( ) 0y ( ) ( ) 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

6 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 a or 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 a or a a, a 0 The Distributive Property If a, b and c are real numbers, then a ( b c) ab a c or ( b c) a ba 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 6

7 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 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 (/2) point from your total. Therefore, if you miss 0 (or more) questions, you will receive zero points. Solve. 4(3x 2) 7 5x Given 4(3x [ 2]) 7 5x. 4(3x ) 4[ 2] 7 5x 2. ( 43) x 4[ 2] 7 5x 3. 2x 4[ 2] 7 5x 4. 2x ( 8) 7 5x 5. 2x [( 8) 7] 5x 6. 2x ( ) 5x 7. 2x ( ) ( 5x) 5x ( 5x) 8. 2x [( ) ( 5x)] 5x ( 5x) 9. 2x [( 5x) ( )] 5x ( 5x) 0. 2x ( 5x) ( ) 5x ( 5x). [ 2 ( 5)] x ( ) 5x ( 5x) 2. 7x ( ) 5x ( 5x) 3. 7

8 7x ( ) 5x [6 ( 5x)] 4. 7x ( ) 5x [ 5x ] 5. 7x ( ) [5x ( 5x)] 6. 7x ( ) [5 ( 5)] x 7. 7x ( ) 0x 8. 7x ( ) x ( ) 20. 7x ( ) 2. 7x [( ) ] 22. 7x x 24. 7x ( 7x ) x x x x 30. x 3. 8