# 2.2 Properties of Algebra

```*
Given an equation, you can …
* Add the same value (or equivalent values) to both sides,
If a = b, then a + 7 = b + 7
* Subtract the same value (or equivalent values) from both sides,
If a = b, then a - 7 = b - 7
* Multiply both sides by the same value (or equivalent values),
If a = b, then 7a = 7b
* Divide both sides by the same value (or equivalent values),
If a = b, then a/7 = b/7
Reflexive Property: a = a
(A value is equal to itself.)
Reflexive Property Examples:: AB  AB or 1  1 or 15  15
Symmetric Property: If a = b, then b = a. (Reverse order)
Symmetric Property Examples: if AB  CD then CD  AB
or if 1  2 then 2  1
*
Substitution Property: if a = b &amp; a = c, then c = b.
1st equation: (a = b): is given to be true;
2nd equation (a = c): indicates a &amp; c are equivalent
3rd equation (c = b): rewrite of 1st equation with c substituted
for a
Example: (Substitution method for simultaneous equations)
x+y=5
(Given)
y = 13 (Establishes that y &amp; 13 are equivalent)
x + 13 = 5 (Substitute 13 for y in 1st equation)
*
Transitive Property : if a = b &amp; b = c, then a = c.
a=c
Note: You can distinguish between Substitution Property
and Transitive Property by the order of the values in the 2
equations.
In proofs you are allowed to use them interchangeably.
Substitution Property: if AB  CD &amp; AB  EF then EF  CD
or
if 1  2 &amp; 1  3 then 3  2
Transitive Property: if AB  CD &amp; CD  EF then AB  EF
or
if 1  2 &amp; 2  3 then 1  3
Prove: If 3x = 7 - .5x then x = 2.
Given: 3x = 7 - .5x
Prove: x = 2
Proof:
Reasons
Statements
1. 3x = 7 - .5x
1. Given
2.
6x = 14 - x
2. Multiplication Property of Equality
3.
7x = 14
4.
x=2
4. Division Property of Equality
Given: m1  m3
Prove: DEG  HEF
Proof:
Statements
1. m1  m3
H
G
1
D
2
3
E
F
Reasons
1. Given
2. m1 m2  m2  m3 2. Add. Property of =
3. m1 m2  mDEG