1. Lesson 24
Algebraic Proofs

2. A proof is an argument that uses logic to show that a conclusion is true
Since you have learned how to solve an algebraic equation, you have already performed a proof (with the exception of giving reasons) These are some of the properties you will use for reasons (page 151)
Property
Example
Addition Prop. Of Equality
If a = b, then a + c = b + c.
Subtraction Prop. Of Equality
If a = b, then a – c = b – c.
Multiplication Prop. Of Equality
If a = b, then ac = bc.
Division Prop. Of Equality
If a = b, then (a/c) = (b/c).
Symmetric Prop. Of Equality
If a = b, then b = a.
Reflexive Prop. Of Equality
a = a
Substitution Prop. Of Equality
If a = 2, then a + 7 =
Transitive Prop. Of Equality
If a = b & b = c, then a = c.

3. Solve & justify each step 5 2𝑥−3 =𝑥+3
Statements
Reasons
5 2𝑥−3 =𝑥+3
10𝑥−15=𝑥+3
9𝑥−15=3
9𝑥=18
𝑥=2
Given
Distributive Property
Subt. Prop. Of Equality
Add. Prop. Of Equality
Div. Prop. Of Equality

4. Solve & justify Alternate Method 5 2𝑥−3 =𝑥+3
Statements
Reasons
5 2𝑥−3 =𝑥+3
10𝑥−15=𝑥+3
−𝑥 −𝑥
9𝑥−15=3
9𝑥=18
𝑥=2
Given
Distributive Property
Subt. Prop. Of Equality
Simplify
Add. Prop. Of Equality
Div. Prop. Of Equality

5. Solve & justify each step 2𝑥+2 3 = 5 6
Statements
Reasons
2𝑥+2 3 = 5 6
6 2𝑥+2 =3(5)
12𝑥+12=15
12𝑥=3
𝑥= 1 4
Given
Cross Multiply
Distributive Property
Subt. Prop. Of Equality
Div. Prop. Of Equality

6. The area of a rectangular garden is 315 square feet
The area of a rectangular garden is 315 square feet. The garden’s length is 𝑥+6 feet and the width is 𝑥 feet. Find its dimensions and justify each step.
Statements
Reasons
𝐴=315, 𝑙= 𝑥+6 , 𝑤=𝑥
𝐴=𝑙𝑤
315= 𝑥+6 𝑥
𝑥+6 𝑥=315
𝑥 2 +6𝑥=315
𝑥 2 +6𝑥−315=0
𝑥+21 𝑥−15 =0
Given
Formula for area of Rect.
Sub. Prop. Of Equality
Sym. Prop. Of Equality
Distr. Prop.
Subt. Prop. Of Equality
Factor

7. The area of a rectangular garden is 315 square feet
The area of a rectangular garden is 315 square feet. The garden’s length is 𝑥+6 feet and the width is 𝑥 feet. Find its dimensions and justify each step.
Statements
Reasons
𝑥+21 𝑥−15 =0
𝑥+21=0 𝑜𝑟 𝑥−15=0
𝑥=−21 𝑜𝑟 𝑥=15
𝑤=15 𝑓𝑡.
𝑙=(15+6) 𝑓𝑡.
𝑙=21 𝑓𝑡.
Why did we ignore -21?
Factor
Zero Product Prop.
Subt./Add. Prop. Of =
Sub. Prop. Of Equality
Simplify
Can’t have neg. dimensions

8. Conclusion/Review Looking forward Lesson 27: 2-Column Proofs
Help for understanding properties
Symmetric (symmetry) is the same on both sides and order doesn’t matter
Reflexive (reflect) is the exact same thing on both sides
Substitution (substitute) is used in place of another
Transitive (transit) is passing from one to another
Looking forward
Lesson 27: 2-Column Proofs
Lesson 31: Flowcharts & Paragraph Proofs
Lesson 45: Intro. To Coordinate Proofs
Lesson 48: Indirect Proofs