Algebra

1. Algebra 3.4 Algebra Properties mbhaub@mpsaz.org

2. Goals • Use properties from algebra. • Use properties to justify statements. Geometry 2.4 Reasoning with Algebra Properties

3. Algebra Properties • Don’t copy all of this down • You have had most before. Copy the ones that are new to you • You need to have them all memorized. • This as well as all presentations are available on-line. Geometry 2.4 Reasoning with Algebra Properties

4. Addition Property Addition Property If a = b, then a + c = b + c. Example x – 12 = 15 x – 12 + 12 = 15 + 12 x = 27 Geometry 2.4 Reasoning with Algebra Properties

5. Subtraction Property Subtraction PropertyIf a = b, then a – c = b – c Example x + 30 = 45 x + 30 – 30 = 45 – 30 x = 15 Geometry 2.4 Reasoning with Algebra Properties

6. Multiplication Property Multiplication Property If a = b, then ac = bc Example Geometry 2.4 Reasoning with Algebra Properties

7. Division Property Division Property Example Geometry 2.4 Reasoning with Algebra Properties

8. Other Properties Reflexive Property For any number a, a = a. Symmetric Property If a = b, then b = a. Transitive Property If a = b and b = c, then a = c Substitution Property If a = b and a = c, then b = c Distributive Property a(b + c) = ab + ac Geometry 2.4 Reasoning with Algebra Properties

9. Identify the Property If 43 = x, then x = 43. Property? Symmetric Geometry 2.4 Reasoning with Algebra Properties

10. Identify the Property If 3x = 12, then 12x = 48 Property? Multiplication Geometry 2.4 Reasoning with Algebra Properties

11. Identify the Property If x = y, and y = 10, then x = 10 Property? Transitive Geometry 2.4 Reasoning with Algebra Properties

12. Identify the Property If x = 12, then x + 2 = 14 Property? Addition Geometry 2.4 Reasoning with Algebra Properties

13. Using a Property Addition Property: If n = 14, then n + 2 = _________ 16 Geometry 2.4 Reasoning with Algebra Properties

14. Using a Property Symmetric: If AB = CD, then CD = ________ AB Geometry 2.4 Reasoning with Algebra Properties

15. Using a Property Transitive: If mA = mB, and mB = mC, then mA = _________. mC Geometry 2.4 Reasoning with Algebra Properties

16. Justification One of the main reasons to study Algebra is to learn how to prove things. The whole business of math is proving things. To prove things in math you must be able to justify everything with legitimate reasons. Our reasons include: postulates, definitions and algebra properties. Geometry 2.4 Reasoning with Algebra Properties

17. Example Solve 3x + 12 = 8x – 18 and write a reason for each step. 3x + 12 = 8x – 18 Given 3x – 8x + 12 – 12 = 8x – 8x – 18 – 12 Subtraction Property –5x = –30 Simplify (combine like terms) –5x/(–5) = –30/(–5) Division Property x = 6 Simplify Geometry 2.4 Reasoning with Algebra Properties

18. Another Example (w/o intermediate steps) Solve x+2(x – 3) = 5x + 2 x + 2(x – 3) = 5x + 2 Given x + 2x – 6 = 5x + 2 Distributive Prop. 3x – 6 = 5x + 2 Simplify 3x = 5x + 8 Addition Prop. –2x = 8 Subtraction Prop. x = – 4 Division Prop. Geometry 2.4 Reasoning with Algebra Properties

19. Before we do a proof… Given Any algebra proof must begin with information that we know is true. This will be given to us as a place to start, so it is called the “given”. There can be one or more givens in a problem. Geometry 2.4 Reasoning with Algebra Properties

20. This is “Proof”. If you feel uncomfortable and confused, that’s normal. Everyone is confused with proof at first. There is only one way to learn proof: PRACTICE. You have to know the properties, postulates and definitions. You must diligently practice by doing the homework every night – NO EXCUSES. You learn by making mistakes. Everyone does. Geometry 2.4 Reasoning with Algebra Properties

21. Proof is essential. Proof is a mandatory part of higher math. If you plan on going to college and/or taking more advanced math you must prove things. Algebra is the place to learn to do this. We will do proofs until the end of the year. Don’t fight it, they are not going to go away. Geometry 2.4 Reasoning with Algebra Properties