1. 2.4 Reasoning with Properties of Algebra

2. Goal 1: Using Properties from Algebra
Pg. 96
Addition property If a = b, then a+c = b+c
Subtraction property If a=b, then a-c=b-c
Multiplication property If a=b, then ac=bc
Division property If a=b, then ac=bc
Reflexive property a=a
Symmetric property If a=b, then b=a
Transitive property If a=b and b=c, then a=c
Substitution property If a=b, then a can be
substituted for b in any
expression or equation.

3. Distributive Property
a (b + c) = ab + ac
Algebra Properties can be used to solve equations
Example:
x + 3 = 7 By subtracting 3 from each side of the equation, you obtain 4.

4. Example 1: Writing Reasons
Solve 5x – 18 = 3x +2
5x – 18 = 3x + 2
2x – 18 = 2
2x = 20
x = 10
Given
Subtraction property
Addition property
Division property

5. Example 2: Writing Reasons
Solve 55z – 3(9z + 12)= -64
55z – 3(9z + 12)= -64
55z – 27z – 36 = -64
28z – 36 = -64
28z = -28
z = -1
Given
Distributiveproperty
Simplify
Addition property
Division property

6. Example 3: Using properties in Real Life
Before exercising, you should find your target heart rate. This is the rate at which you achieve an effective workout while not placing too much strain on your heart. Your target heart rate (r) –in beats per minute can be determined from your age (a)—in years using the equation a = 220 – 10/7 r.

7. Find the following:
Solve the equation for r and write a reason for each step.
Use the result to find the target heart rate for a 16-year old.
Find the target rate for the following ages: 20, 30, 40, 50, and 60. What happens to the target heart rate as a person gets older?

8. a. a = 220 – 10/7 r a = 220 – 10/7r a + 10/7 r = 220 10/7r = 220 – a
Given
Addition property
Subtraction property
Multiplication property

9. b. Using a = 16, the target rate is:
The target rate for a 16 year old is about 143 beats per minute
Given
Substitute 16 for a
Simplify

10. c.
Age
Rate 20
140 30
133 40
126 50
119 60
112
From the table, the target heart rate appears to decrease as the person gets older.

11. Goal 2: Using Properties of Length and Measure
Segment Length
Reflexive: AB = AB
Symmetric: If AB=CD, then CD=BA
Transitive: If AB=CD and CD=EF, then AB=EF.

12. Angle Measure Properties
Reflexive: mA=mA
Symmetric: If mA = mB, then mB = mA.
Transitive: If mA = mB and mB = mC, then mA = mC.

13. Example 4: Using properties of length
AB = CD
AB + BC = BC + CD
AC = AB + BC
BD = BC + CD
AC = BD
Given
Addition property
Segment addition postulate
Substitution property

14. Example 5: Using properties of measure
m1 + m  2 = 66
m  1 + m  2+m  3 = 99 
66  + m  3 = 99
m  3 = 33 
m  3 = m  1, m  1 = m  4
m  3 = m  4
m  4 = 33 
Given
Substitution
Subtraction
Transitive