2.2.1 Algebraic Proof
Key Objectives • Review properties of equality and use them to write algebraic proofs. • Identify properties of equality and congruence. Key Terms • A proof is an argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true. Theorems, Postulates, Corollaries, and Properties • Properties of Equality There are eight Properties of Equality: Addition, Subtraction, Multiplication, Division, Reflexive, Symmetric, Transitive, and Substitution. • Distributive Property a ( b + c ) = ab + ac and a ( b − c ) = ab − ac • Properties of Congruence There are three Properties of Congruence: Reflexive, Symmetric, and Transitive.
The eight Properties of Equality are listed above. The first four, Addition, Subtraction, Multiplication, and Division are commonly used when solving algebraic equations. Basically, these four properties show that an equation can be added, subtracted, multiplied, or divided by any number, as long as that same number and operation is applied to both sides of the equation. The Substitution Property of Equality is commonly used when evaluating algebraic expressions and when checking that the solution to an algebraic equation is correct. Example 1 Solving an Equation in Algebra To solve an algebraic equation, use the Properties of Equality and manipulate the equation so that the variable will be isolated on one side of the equation. Remember to apply any operation to both sides of the equation. Solving an equation and justifying the steps using properties or definitions is an example of an algebraic proof.
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