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Reason Using Properties from Algebra 2.5 Looking back at algebra, we used different properties of real numbers to solve an equation. These properties include: Algebraic Properties of Equality (Let a, b, and c be real numbers) Addition Property If a=b, then a+c=b+c Subtraction Property If a=b, then a-c=b-c Multiplication Property If a=b, then ac=bc Division Property If a=b, then Substitution Property If a=b, then a can be substituted for b in any equation or expression π π = π π Example: Solve 2π₯ + 5 = 20 β 3π₯ and write a reason for each step. 2π₯ + 5 = 20 β 3π₯ 5π₯ + 5 = 20 5π₯ = 15 π₯=3 Given Addition Property of Equality Subtraction Property of Equality Division Property of Equality We can also use the Distributive Property: π(π + π) = ππ + ππ where a, b, and c are real numbers. Example: Solve β4(11π₯ + 2) = 80 and write a reason for each step. β4(11π₯ + 2) = 80 (11π₯ + 2) = β20 11π₯ = β22 π₯ = β2 Properties of Equality (true for all real numbers) Reflexive Property of Equality ο· Real Numbers: For any real number π, π = π ο· Segment Length: For any segment Μ Μ Μ Μ π΄π΅, π΄π΅ = π΄π΅ ο· Angle Measure: For any angle β A, πβ A = mβ A Symmetric Property of Equality ο· Real Numbers: For any real numbers π and π, if π = π, then π = π ο· Segment Length: For any segments Μ Μ Μ Μ π΄π΅ and Μ Μ Μ Μ πΆπ·, if π΄π΅ = πΆπ·, then πΆπ· = π΄π΅ ο· Angle Measure: For any angles β A and β B, if πβ A = mβ B, then πβ B = mβ A Transitive Property of Equality ο· Real Numbers: For any real numbers π, π, and π, if π = π and π = π, then π = π ο· Segment Length: For any segments Μ Μ Μ Μ π΄π΅, Μ Μ Μ Μ πΆπ·, and Μ Μ Μ Μ πΈπΉ , if π΄π΅ = πΆπ· and πΆπ· = πΈπΉ, then π΄π΅ = πΈπΉ ο· Angle Measure: For any angles β A, β B, and β C, if πβ A = mβ B and πβ B = mβ C, then πβ A = mβ C Example: (Use the properties of equality) You are designing a logo to sell daffodils. Use the information given. Determine whether πβ EBA = mβ DBC πβ 1 = πβ 3 πβ πΈπ΅π΄ = πβ 2 + πβ 3 πβ πΈπ΅π΄ = πβ 2 + πβ 1 πβ 1 + πβ 2 = πβ π·π΅πΆ πβ πΈπ΅π΄ = πβ π·π΅πΆ