Algebraic proofs A proof is an argument that uses logic to show that a conclusion is true. Every time you solved an equation in Algebra you were performing a proof since an algebraic proof uses properties of equality to solve an equation.

Properties of Equality Addition prop of Equality Subtraction prop of equality Multiplication prop of equality Division prop of equality Symmetric prop of equality Reflexive prop of equality Transitive prop of equality Substitution prop of equality Example If a=b, then a+ c = b +c If a=b, then a-c= b-c If a = b, then ac = bc If a= b, c=0, then a/c=b/c If a= b, then b=a a=a If a= b and b= c, then a =c If a= b, then b can be substituted for a in any expression

Algebraic Proof An algebraic proof shows step by step how a problem is solved. Whenever a step requires that you perform basic mathematical operations on a single side of an equation( like add, subtract, multiply or divide), the step is justified by the term, “simplify”

Most proofs begin by presenting the facts of a problem. The first line restates what you have been told, and is justified as "Given"

Writing an algebraic proof Solve this equation. Justify each step Writing an algebraic proof Solve this equation . Justify each step. 2(x+1) = x+9 2(x+1)= x+9 2x+2= x+9 2x+2-2= x+9-2 2x= x+7 2x-x= x+7-x X=7 Given Distributive property Subtr prop of equality Simplify simplify

3x-1 = 2x+3 5 3 3x-1 = 2x+3 5 3 15(3x-1)=15(2x+3) 5 3 5 3 9x-3 = 10x +15 9x -3 +3= 10x +15 +3 9x = 10x + 18 9x -10x =10x +18 -10x -x = 18 -x(-1) = 18(-1) X = -18 Given Mult prop of equality Associative prop of mult Distributive prop Add prop of equality Simplify Subtr prop of equality simplify

Give justification for each step. 2 2 a = -4 1) 2) 3) 4) 5) 6)