Algebra

3.4 Algebra Properties

mbhaub@mpsaz.org

April 10, 2023Geometry 2.4 Reasoning with Algebra

Properties 2

Goals Use properties from algebra. Use properties to justify

statements.

April 10, 2023Geometry 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.

April 10, 2023Geometry 2.4 Reasoning with Algebra

Properties 4

Addition PropertyAddition Property

If a = b, then a + c = b + c.

Example

x – 12 = 15

x – 12 + 12 = 15 + 12

x = 27

April 10, 2023Geometry 2.4 Reasoning with Algebra

Properties 5

Subtraction PropertySubtraction PropertyIf a = b, then a – c = b – c

Example

x + 30 = 45

x + 30 – 30 = 45 – 30

x = 15

April 10, 2023Geometry 2.4 Reasoning with Algebra

Properties 6

Multiplication PropertyMultiplication Property

If a = b, then ac = bc

Example 2 123

3 2 3 (12)2 3 2

18

x

x

x

April 10, 2023Geometry 2.4 Reasoning with Algebra

Properties 7

Division Property

0)(c cb

ca

then b, a If .

Division Property

Example

2.012

4.2

12

12

4.212

x

x

x

April 10, 2023Geometry 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

April 10, 2023Geometry 2.4 Reasoning with Algebra

Properties 9

Identify the Property

If 43 = x, then x = 43.

Property?

Symmetric

April 10, 2023Geometry 2.4 Reasoning with Algebra

Properties 10

Identify the Property

If 3x = 12, then 12x = 48

Property?

Multiplication

April 10, 2023Geometry 2.4 Reasoning with Algebra

Properties 11

Identify the Property

If x = y, and y = 10, then x = 10

Property?

Transitive

April 10, 2023Geometry 2.4 Reasoning with Algebra

Properties 12

Identify the Property

If x = 12, then x + 2 = 14

Property?

Addition

April 10, 2023Geometry 2.4 Reasoning with Algebra

Properties 13

Using a Property

Addition Property:

If n = 14, then n + 2 = _________16

April 10, 2023Geometry 2.4 Reasoning with Algebra

Properties 14

Using a Property

Symmetric:

If AB = CD, then CD = ________ AB

April 10, 2023Geometry 2.4 Reasoning with Algebra

Properties 15

Using a Property

Transitive:

If mA = mB, and mB = mC,

then mA = _________.mC

April 10, 2023Geometry 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.

April 10, 2023Geometry 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

–5x = –30

Subtraction Property

x = 6

Division Property

3x – 8x + 12 – 12 = 8x – 8x – 18 – 12

Simplify (combine like terms)–5x/(–5) = –30/(–

5)

Simplify

April 10, 2023Geometry 2.4 Reasoning with Algebra

Properties 18

Another Example (w/o intermediate steps)

Solve x+2(x – 3) = 5x + 2x + 2(x – 3) = 5x + 2x + 2x – 6 = 5x + 2

3x – 6 = 5x + 2

3x = 5x + 8

–2x = 8

x = – 4

Given

Distributive Prop.

Simplify

Addition Prop.

Subtraction Prop.

Division Prop.

April 10, 2023Geometry 2.4 Reasoning with Algebra

Properties 19

Before we do a proof…

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.

April 10, 2023Geometry 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.

April 10, 2023Geometry 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.