Solving equations using properties The story of SIMPLIFYING and THEN SOLVING

WHY PROPERTIES? Math is about RULES—this standardizes our process so we can advance scientifically and technologically as a planet. The fancy name for these RULES is PROPERTIES. We have one set of properties for SIMPLIFYING and one set of properties for SOLVING

Properties used to SIMPLIFY expressions COMMUTATIVE PROPERTY Adding or Multiplying in any order 3+4=4+3 ASSOCIATIVE PROPERTY Grouping to make adding or mult. easier OR TO COLLECT LIKE TERMS CORRECTLY (3X + 4)+ 8= 3X + (4+8) DISTRIBUTIVE PROPERTY Multiplying by all terms inside parenthesis 3(4x+2)=12x+6

Examples of Simplifying 4x + 2(2x +3) Given 4x + 4x + 6Distributive Property (4x +4x) +6Assoc. Prop. 8x + 6Simplify (3x +5)Given 8 + 6x +10Dist xCommutative Prop. (8+10)+6xAssoc. Prop xSimplify

Partner QUESTIONS What are the three most common properties used when SIMPLIFYING expressions? Explain the Commutative Property Describe the Associate Property Give an example of the Distributive Property Simplify 4x + 8 – 7x -3 Simplify 2x – 10 – 12x + 14 Simplify 4 – 3(2x + 5) Simplify 5x – (3x – 4)

Properties used to SOLVE equations Once you are SOLVING EQUATIONS which you RECOGNIZE because you have an EQUALS sign you will use other properties EQUALITY PROPERTY Most basic and common—adding, subtracting, multiplying or dividing the SAME number from BOTH sides of the EQUALS sign 3x+2=

INVERSE PROPERTY You pick which operation to use with the equality property based on the INVERSE of the number you’d like to eliminate The INVERSE of + 3 is – 3 This INVERSE is FORMALLY known as ADDITIVE INVERSE The INVERSE of *3 is divide by 3 This INVERSE is FORMALLY known as the multiplicative inverse of 3 is 1/3

IDENTITY PROPERTY You use the INVERSE to find your IDENTITY PROPERTY so you can eliminate terms and SOLVE for the variable. 3x + 2 = 14Given Equality Property 3x + 0 = 12Additive Inverse 3x=12 Identity Property /3 /3 Equality Property 1*x =4 Multiplicative Inverse x=4Identity Property

Using Simplifying and Solving Together SO in Mutistep Equations FIRST you simplify THEN you SOLVE 8 – 4(x+1)=8Given 8 - 4x – 4 = 8Distributive Property 8 – 4 – 4x = 8Commutive Property (8 – 4) – 4x = 8Associative Property 4 – 4x = 8Simplify These are the ways you use the SIMPLIFYING PROPERTIES

Properties we use to SOLVE equations Name the properties we use to solve equations Explain the Equality Property Define the Inverse Property Give an example of the Identity Property

NOW TO SOLVE 4 – 4x = 8Finished simplifying Equality Property 0 – 4x = 4Inverse -4x= 4Identity /-4 /-4 Equality 1*x=-1 Inverse X=-1 Identity

Now write a summary of everything I just showed you