Notes: ALGEBRAIC PROPERTIES
Content Objective: I will be able to use algebraic properties to prove logical
arguments.
PROPERTIES OF EQUALITY
PROPERTY
ALGEBRAIC EXAMPLE
EXAMPLE
If x – 3 = 6, then x = ___9______
Addition Property
If a = b, then a + c = b + c
If 4x – 12
= -3x + 6,
then 4x – 12 + 12 = -3x + 6 +12
Subtraction Property
If a = b, then a – c = b - c
If 2x +12 =20, then 2x = __8____
Multiplication
Property
If a = b, then a · c = b ·c
If
Division Property
If a = b, then
a b

c c
PROPERTY
ALGEBRAIC EXAMPLE
Reflexive Property
a=a
x
 2 , then x = ___-8_____
4
If -6x = 20 , then x =___-10/3_____
EXAMPLE
a) 3 = ____3_______
a) If _c =d___, then ___d = c___
Symmetric Property
If a = b, then b = a
b) If 3 = x, then __x = 3___
Transitive Property
If a = b, b = c, then a = c
a) If __x = 3_____ , __3 = 6/2_____ then
___x = 6/2_____
b) If 2 = x, x = a, then _2 = a___
____
Geometry Unit 1 - Essentials of Geometry
Page 1
Notes: Algebraic Properties
a) a(b + c) = ___ab + ac____
Distributive Property
b) If -2(4x - 3), then -8x + 6
___________________
a) If x = 3 and x + 8, ____3 + 8______
Substitution Property
b) If x = -2 and 2x + 8, then _2(-2) + 8____
EXAMPLE 1: Use the property to complete each statement.
1. Addition Property of Equality: A – 15
= 3
A – 15 + _15_ = 3 + 15__ .
2. Distributive Property of Equality: 3(2x – 1) = __6x - 3___.
QUICK CHECK:
3. Symmetric Property of Equality: If 3 = x, then x = 3
4. Transitive Property of Equality: If A = B and B = C, then
A=C
5. Substitution Property of Equality: If Y = 2, then
R + Y = _R + 2__
Geometry Unit 1 - Essentials of Geometry
Page 2
Notes: Algebraic Properties
EXAMPLE 2: The following equation has been solved, justify each step.
-2(3x – 1) = 180 Statements
-2(3x – 1) =180
-6x + 2 = 180
Reasons
Given
Distributive Property
-6x + 2 - 2 = 180 - 2
Addition Property of Equality
-6x = 178
Simplify.
 6 x 178

6
6
x =  29
2
3
Division Property of equality
Simplify.
QUICK CHECK: The following equation has been solved, justify each step.
3x – 5 = 90
Statements
3x – 5 = 90
3x – 5 + 5 = 90 + 5
3x = 95
3 x 95

3
3
x
95
3
Geometry Unit 1 - Essentials of Geometry
Reasons
Given
Addition Property of Equality
Simplify by combining like terms.
Division Property of Equality
Simplify by combining like terms.
Page 3
Notes: Algebraic Properties
EXAMPLE 3: The following equation has been solved, justify each statement.
1
x  5  2x  7
2
Statements
Reasons
1
x  5  2x  7
2
Given
1
x  5  5  2x  7  5
2
1
x  2 x  12
2
1
2( x )  2(2 x  12)
2
x  4 x  24
Addition Property of Equality
x  4 x  4 x  4 x  24
 3 x  24
 3 x 24

3 3
x  8
Simplify by combining like terms
Multiplication Property of Equality
Simplify
Subtraction Property of Equality
Simplify by combining like terms
Division Property of Equality
Simplify
QUICK CHECK: The following equation has been solved, justify each
statement. 2 x  15  19  2 x
Statements
Reasons
2 x  15  19  2 x
Given
2 x  15  15  19  15  2 x
Subtraction Property of Equality
2x  4  2x
Simplify by combining like terms
2x  2x  4  2x  2x
Addition Property of Equality
4x  4
Simplify by combining like terms
4x 4

4 4
Division Property of Equality
x 1
Geometry Unit 1 - Essentials of Geometry
Simplify
Page 4
Notes: Algebraic Properties
EXAMPLE 4: The following equation has been solved, justify each step.
8x + 4(2+x) = 180
Statements
8x  8  4x  180
12x  8  180
12x  8  8  180  8
12 x  172
12 x 172

12
12
1
x  14
3
Reasons
Distributive Property
Simplify by combining like terms
Subtraction Property of Equality
Simplify by combining like terms
Division Property of Equality
Simplify
QUICK CHECK: Solve the following equation. Justify each step.
1
x  5  90
2
Statements
½ x – 5 = 90
Reasons
Given
Addition Property of Equality
½ x – 5 + 5 = 90 + 5
½ x = 95
2( ½ x) = 2(95)
Simplify by combining like
terms
Multiplication Property of
Equality
Simplify.
X = 190
Geometry Unit 1 - Essentials of Geometry
Page 5
Notes: Algebraic Properties
EXAMPLE 5: Fill in any missing statements or reasons to solve the equation.
Statements
Reasons
x  10
Given
x 1 
2
x  10
 2( x  1) 
(2)
2
-2x +2 = x - 10
Multiplication Property of Equality
Distributive Property
 2x  2x  2  x  2x  10
2 = 3x - 10
Addition Property of Equality
Simplify by combining like terms
2 + 10 = 3x – 10 + 10
12 = 3x
Addition Property of Equality
Simplify like terms
Division Property of Equality
12 3x

3
3
4=x
Simplify
1
3
QUICK CHECK: If 2 x  7  x  2, prove x  3
Statements
2x  7 
1
x2
3
Reasons
Given
2x -7 + 7 = 1/3 x – 2 + 7
1
2x = /3 x + 5
2x - 1/3 x = 1/3 x - 1/3 x + 5
5
3
/ 3x = 5
Addition Property of Equality
Simplify by combining like terms
Subtraction Property of Equality
Simplify by combining like terms
/5 (5/3 )x = 3/5 (5)
Multiplication Property of Equality
X=3
Simplify
Geometry Unit 1 - Essentials of Geometry
Page 6