Algebra Period 4

EQUATIONS Sections 3-1 to 3-3

We'll be using our  EQUATION BALANCING PROPERTIES OF EQUALITY:
There are 4 of these.
Whatever YOU DO TO BALANCE an equation, 
that operation is the property of equality that was used.
 
So if you have x + 3 = 10, you used the SUBTRACTION PROPERTY OF EQUALITY because you need to SUBTRACT 3 from each side equally. "- prop ="
If you have x - 3 = 10, you used the ADDITION PROPERTY OF EQUALITY because you need to ADD 3 from each side equally. " + prop ="
If you have 3x = 10, you used the DIVISION PROPERTY OF EQUALITY because you need to DIVIDE each side equally by 3. " / prop ="
If you have x/3 = 10, you used the MULTIPLICATION PROPERTY OF EQUALITY because you need to MULTIPLY each side equally by 3. "x prop ="
 
SOMETIMES, WE SAY THERE ARE ONLY 2 BALANCING PROPERTIES OF EQUALITY
CAN YOU GUESS WHICH 2 ARE "DROPPED OUT"?
Since we say we never subtract and we really never divide, it's those 2.
GOING BACK TO OUR PREVIOUS EXAMPLES:
If you have x + 3 = 10, you could say that we ADDED -3 to each side equally; therefore, we used the ADDITION (not subtraction) PROPERTY.
If you have 3x = 10, you could say that we MULTIPLIED each side equally by 1/3; therefore, we used the MULTIPLICATION (not division) PROPERTY. 
(We always multiply by the MULTIPLICATIVE INVERSE).
 

REVIEW OF SIMPLE EQUATIONS: One and Two Step.
What's the 
GOAL? Determine the value of the variable

HOW? Isolate the variable (get it alone on one side of equation)

WHAT to do? Use inverse (opposite) operations to "get rid" of everything on the side with the variable

WHAT SHOULD MY FOCUS BE WHEN EQUATIONS GET COMPLICATED?
Always focus on the variable(s) first!!!!!!!



TWO STEP EQUATIONS:

1. Use the ADDITION/SUBTRACTION PROPERTIES OF EQUALITY first
(get rid of addition or subtraction)

2. Use the MULTIPLICATION/DIVISION PROPERTIES OF EQUALITY second
(get rid of mult/division)
 
IDENTITY PROPERTIES AND INVERSE PROPERTIES  are also used to justify solving equations!
When you have a one-step equation such as x + 5 = 12, you ADD -5 (or just subtract 5) from each side equally. The reason you chose -5 is that it was the ADDITIVE INVERSE of 5.  The reason the +5 then "disappears" is due to the IDENTITY PROPERTY OF ADDITION. Since +5 + (-5) = 0, it's not necessary to bring down the 0 in the equation.
 
JUSTIFYING A SIMPLE ONE STEP:
 
x + 5 = 12     GIVEN
- 5   -5     Subtraction Prop =
X + 0  = 7     Additive Inverse Prop
x       =  7   Identity of Addition


COLLECTING TERMS FIRST:

Sometimes, you will have to COLLECT LIKE TERMS ON THE SAME SIDE OF THE EQUATION before balancing



FORMAL CHECK:
 Follow these three steps---
1. Rewrite original equation

2. Substitute your solution and question mark over the equal sign

3. Do the math and check it!

2- Steps with Distributive Property
USUALLY you want to distribute first!!
(Unless the factor outside the ( ) can be divided out of BOTH SIDES PERFECTLY!!

5y -2(2y +8) = 16
5y -4y -16 = 16 {distribute accurately}
y - 16 = 16 { collect like terms}
y = 32 { solve by adding 16 to both sides!!}

-3(4 +3x) = -9
Wait you could distribute but if you divide both sides by -3
{the -3 goes into BOTH SIDES PERFECTLY}
you have 4 + 3x = 3
3x = -1 { subtract 4 from both sides}
x = -1/3 { Divide both sides by 3}

What if you have variables on BOTH SIDES of the equation?

Simplify each side of the equation first
THEN use the additive inverse property to move the variables to the other side, Usually we try to move the smaller coefficient to the larger because some times that avoids negative coefficients. But it is not always the case.

3y - 10 - y = -10y + 12
combine like terms
2y - 10 = -10y +12
In this case I would add 10y to both sides to get rid of a negative coefficient
12y - 10 = 12
Now add 10 to both sides
12y = 22
Now divide both sides by 12
y = 22/12 or y = 11/6