Identity and Equality Properties

Identity and Equality Properties Properties refer to rules that indicate a standard procedure or method to be followed. A proof is a demonstration of the truth of a statement in mathematics. Properties or rules in mathematics are the result from testing the truth or validity of something by experiment or trial to establish a proof. Therefore every mathematical problem from the easiest to the more complex can be solved by following step by step procedures that are identified as mathematical properties.

Identity and Equality Properties Identity Properties Additive Identity Property For any real number a, a + 0 = 0 + a = a. The sum of any number and zero is equal to that number. The number zero is called the additive identity. Example If a = 5 then 5 + 0 = 0 + 5 = 5

Identity and Equality Properties Identity Properties Multiplicative identity Property For any number a, a  1 = 1  a = a. The product of any number and one is equal to that number. The number one is called the multiplicative identity. Example If a = 6 then 6  1 = 1  6 = 6

Identity and Equality Properties Identity Properties For any integer a, Additive Inverse Property Two numbers whose sum is 0 are called additive inverses. Zero has no inverse because 0 plus 0 is 0. Example

Identity and Equality Properties Identity Properties Multiplicative Inverse Property Two numbers whose product is 1 are called multiplicative inverses or reciprocals. Zero has no reciprocal because any number times 0 is 0. Example

Identity and Equality Properties Equality Properties allow you to compute with expressions on both sides of an equation by performing identical operations on both sides of the equation. This creates a balance to the mathematical problem and allows you to keep the equation true and thus be referred to as a property. The basic rules to solving equations is based on these properties. Whatever you do to one side of an equation; You must perform the same operation(s) with the same number or expression on the other side of the equals sign.

Identity and Equality Properties Substitution Property of Equality If a = b, then a may be replaced by b in any expression. The substitution property of equality says that a quantity may be substituted by its equal in any expression. Many mathematical statements and algebraic properties are written in if-then form when describing the rule(s) or giving an example. The hypothesis is the part following if, and the conclusion is the part following then. Example If 8 + 4 = 7 + 5; since 8 + 4 = 12 or 7 + 5 = 12; Then we can substitute either simplification into the original mathematical statement.

Identity and Equality Properties Addition Property of Equality If a = b, then a + c = b + c or a – c = b - c The addition property of equality says that if you add or subtract equal quantities to each side of the equation you get equal quantities. Many mathematical statements and algebraic properties are written in if-then form when describing the rule(s) or giving an example. The hypothesis is the part following if, and the conclusion is the part following then. Example If 6 = 6 ; then 6 +3 = 6 + 3 or 6 – 3 = 6 - 3 If a = b ; then a + c = b + c or a – c = b - c

Identity and Equality Properties Multiplication Property of Equality   The multiplication property of equality says if you multiply or divide (multiply by a fraction) equal quantities to each side of the equation you get equal quantities.   Example

Identity and Equality Properties Associative Property for Addition and Multiplication If a, b, & c are real numbers, then a + (b + c) = (a + b) + c or a(bc) = (ab)c The associative property states when adding, the order the numbers are grouped does not change the sum. The associative property states when multiplying, the order the numbers are grouped does not change the product 7 + (3 + 2) = (7 + 3) + 2 = 12 7 · (3 · 2) = (7 · 3) · 2 = 42 Example

Identity and Equality Properties If a & b are real numbers, then a +b = b +a or a(b) = b(a) Commutative Property for Multiplication and Addition The commutative property states when adding, the order the numbers are added does not change the sum. The commutative property states when multiplying, the order the numbers are multiplied does not change the product 7 + 3 = 3 + 7 = 10 7 (3) = 3 (7) = 21 Example

Identity and Equality Properties Distributive Property If a, b, & c are real numbers, then a (b + c) = ab + ac The distributive property states when multiplying a sum by a real number, the number outside the parentheses is multiplied by each term inside the parentheses then added. 7 (3 + 2) = (7)(3) + (7)(2) = 21 + 14 = 35 Example

Distributive Property     Given   Distributive Property   Simplify Additive Inverse/Addition Property of Equality     Additive Identity/Simplify   Multiplicative Inverse/ Multiplication Property of Equality   Multiplicative Identity/Simplify

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