Download 2.5 – Reasoning Using Properties of Algebra

2.5 – Reasoning Using Properties of Algebra
Name
Property of Equality
Addition Property
If a = b, then a + c = b + c
Subtraction Property
If a = b, then a – c = b – c
Multiplication
Property
Division Property
If a = b, then ac = bc
Reflexive Property
For any real #, 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/c = b/c
If a = b, then b can be substituted in for a
PROOFS!
All proofs start and end the same. It is very
helpful to have a picture to refer to. We are
given information and then are told to prove
something.
Format of Proofs
Given:
Prove:
Picture
Statements
Reasons
1. What you are given 1. Given
2.
2.
3.
3.
… To Prove
…
Use the property to complete the statement.
2. Addition Property of Equality:
TU + 20
If RS = TU, then RS + 20 = ______________.
3.Multiplication Property of Equality:
If m1 = m2, then 3m1 = ____________.
3m2
4.Substitution Property of Equality:
= 100
If a = 20, then 5a = 5(20)
__________.
5.Reflexive Property of Equality:
x
If x is a real number, then x = _____.
6.Symmetric Property of Equality:
If AB = CD, then CD = __________.
AB
7.Transitive Property of Equality:
If mE = mF and mF = mG, then
________________.
mE = mG
Complete the logical argument by writing a reason for each step.
Statements
1.
8x – 34 = 6
Reasons
1.
given
__________________
2.
8x = 40
2.
Addition Prop
__________________
3.
x=5
3.
Division Prop
__________________
Complete the logical argument by writing a reason for each step.
Statements
Reasons
1.
4x – 7 = 6x + 7
1.
given
__________________
2.
-2x – 7 = 7
2.
__________________
Subtraction Prop
3.
-2x = 14
3.
Addition Prop
__________________
4.
x = -7
4.
Division Prop
__________________
Complete the logical argument by writing a reason for each step.
Statements
Reasons
1.
5(x – 3) = 4(x + 2)
1.
given
___________________
2.
5x – 15 = 4x + 8
2.
Distributive Prop
__________________
3.
4.
x – 15 = 8
x = 23
3. ___________________
Subtraction Prop
4.
Addition Prop
___________________
9. Solve the equation. Write a reason for each step
4x – 7 = 6x + 7
Statements
Reasons
1. 4x – 7 = 6x + 7
1.
Given
– 7 = 2x + 7
2.
Subtraction Prop.
2.
3.
– 14 = 2x
3.
Subtraction Prop.
4.
–7 = x
4.
Division Prop.
10. Solve the equation for y. Write a reason for each step.
12x – 3y = 30
Statements
Reasons
1. 12x – 3y = 30
1. Given
2. –3y = -12x + 30
2. Subtraction Prop.
3.
y = 4x – 10
3. Division Prop.
Complete the proof by writing a reason for each step.
GIVEN: AL = SK
PROVE: AS = LK
Statements
Reasons
1.
AL = SK
1.
given
___________________
2.
LS = LS
2.
Reflexive Prop
___________________
3. AL + LS = SK + LS
3.
Addition Prop
__________________
4.
AL + LS = AS
Addition Prop
4. Segment
__________________
5.
SK + LS = LK
Addition Prop
5. Segment
___________________
6.
AS = LK
6.
Substitution Prop
___________________
Steps for Proofs:
1. Start with given and mark pictures
2. What do the givens tell us?
3. What else do we know by looking at the
picture?
4. End the proof with what we are trying to prove