Lesson 24 Algebraic Proofs

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 = 2 + 7. Transitive Prop. Of Equality If a = b & b = c, then a = c.

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

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 +15 +15 9𝑥=18 9 9 𝑥=2 Given Distributive Property Subt. Prop. Of Equality Simplify Add. Prop. Of Equality Div. Prop. Of Equality

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

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

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

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