Pre Algebra Period 1

Simplifying Variable Expressions 2-3

Review of Algebraic terminology:

In the expression, 3y + 5

3 is the coefficient (number attached to variable - remember, "co" means to go along with)

y is the variable

5 is the constant (number not attached to variable)

terms are separated by ADDITION ONLY!



COMBINING LIKE TERMS:

1. Same variable (or no variable)

2. Same power
You can combine by addition or subtraction LIKE TERMS.

You cannot combine UNLIKE TERMS.


EX:
3a + 4a = 7a
 BUT

3a + 4b = 3a + 4b

3a + 4a² = 3a + 4a²


YOU SHOULD ALWAYS COMBINE LIKE TERMS BEFORE YOU EVALUATE!

IT'S MUCH SIMPLER!

-25a + 5a - (-10a) when a = -14

First combine like terms: -10a

Then plug in for a = -14:
-10(-14) = 140

VARIABLES AND EQUATIONS: Section 2-4

Is a given number a solution to an equation?

Substitute and evaluate to see.

Use the set signs { } and a ? over the equal sign as we did in class

9k = 10 - k for k = 1


9(1) = 10 - (1)

9 = 9 checks



Numeric Equations are either True or False

Equations with variables are OPEN (don't know if they are True or False)

Algebra Period 4

Chapter 3-7 Formulas


Some formulas you should already know (if you don't, it's time to memorize them!)

d = rt

p = 2l + 2w
in a rectangle

A = lw or bh

THIS LESSON IS NOT ON MEMORIZING, BUT WHAT YOU CAN DO AFTER YOU MEMORIZE THESE FORMULAS.

FORMULAS = special equations that have KNOWN relationships


Example: d = rt

Sometimes, you want to solve for distance, but other times you need the time or the rate.

You can manipulate the variables by balancing the equation until you solve for the wanted variable.

In the formula, d = rt, to solve for rate, divide both sides by t

To solve for time, divide both sides by r

Don't think of this as 3 different equations! 

Just learn the main one and use that to solve for what you need!


Example:
Solve for l:

P = 2l + 2w

Subtact 2w for both sides:
P - 2w = 2l

Divide by 2 on each side:
(1/2)(P - 2w) = l.


When the variable is in the denominator, it's usually easiest to use cross products:

Solve for x:


(3z)/x = 4/y
     
Cross products:
3yz = 4x



Divide both sides by 4:

x = 3yz/4
 
If you have a fraction, you can simply use the multiplicative inverse:


Solve for b:

3ab = 7c
4           


Either use cross products or multiply each side by 4/3a:

b = 7c (4/3a) = 28c/3a


Why can't you use cross products for the following?

Solve for f:

d = 2a/c + 2b/f
 

Math 6H Period 3, 6 & 7

Solving Equations & Inequalities 2-4 & 2-5
Any value of the variable that makes the equation or inequality a true sentence is called a solution. We solve an equation or inequality when we find ALL its solutions.

Remember when you solved the following:
? + 4 = 12.
Your teacher would ask you what number replaced the ?
We are going to learn how to solve equations and inequalities using Algebraic terms-- we will solve using inverse operations and we will justify our steps.
x - 4 = 12
add 4 TO BOTH SIDES of the equation using
the addition property of equality ( + prop =)
so
x - 4 = 12
+ 4 = + 4
x + 0 = 16 but the Identity Property of addition ( ID+) allows us to write
x = 16

Similarly
15n = 60
can be read as "fifteen times n equals sixty"
What undoes multiplication? What is the inverse operation of multiplication?
DIVISION
so we will divide both sides by 15 to isolate the variable.
15n = 60
15 = 15
using the Division Property of Equality ÷prop +
Then we get
1n = 4
but the Identity Property of Multiplication ( ID X) lets us write
n = 4
See the Class Notes Solving One-Step Equation ( Yellow Sheet) for more examples. Make sure to glue that into your spiral notebook.



GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation

Here are the steps and justifications (reasons)
1. focus on the side where the variable is and focus specifically on what is in the way of the variable being by itself ( isolated)
2. What is the operation the variable is doing with that number in its way?
3. Get rid of that number by using the opposite ( inverse) operation
*Use + if there is a subtraction problem
*Use - if there is an addition problem
*Use x if there is a division problem
*Use ÷ if there is a multiplication problem

GOLDEN RULE OF EQUATIONS; DO UNTO ONE SIDE OF THE EQUATION WHATEVER YOU DO TO THE OTHER!!

4. Justification: You have just used one of the PROPERTIES OF EQUALITY
which one?
that's easy-- Whatever operation YOU USED to balance both sides that's the property of equality
We used:
" +prop= " to represent Addition Property of Equality
" -prop= " to represent Subtraction Property of Equality
" xprop= " to represent Multiplication Property of Equality
" ÷prop= " to represent Division Property of Equality

5. You should now have the variable all alone ( isolated) on one side of the equal sign.

6. Justification: Why is the variable alone?
For + and - equations you used the Identity Property of Addition (ID+) which simply means that you don't bring down the ZERO because you add zero to anything-- it doesn't change anything... [Note: there is no ID of subtraction]

For x and ÷ equations, you used the Identity Property of Multiplication (IDx) which simply means that you don't bring down the ONE because when you multiply by one it doesn't change anything [NOTE: there is no ID of division]

7. Put answer in the final form of x = ____and box this in.

Pre Algebra Period 1

EQUATIONS & INEQUALITIES: Properties 2-1




WHAT ARE PROPERTIES? (Why are they your friends?)

You can count on properties. They always work.
There are 0 COUNTEREXAMPLES!

COUNTEREXAMPLE = an example that shows that something does not work
(counters what you have said)
Because you can count on them, you can use them to JUSTIFY what you do.

JUSTIFY = a reason for doing what you did


PROPERTIES ARE EXCEPTIONS TO AUNT SALLY:

Commutative (order) Property
a + b = b + a

ab = ba

3 + 5 = 5 + 3

3 (5) = 5 (3)

(you can HEAR the change in order!)

Aunt Sally says that you always need to go left to right, but Commutative says not necessary if you have all multiplication or all addition.



Associative (groupings) Property
a + b + c = a + (b + c)

abc = a(bc)

(3 + 2) + 8 = 3 + (2 + 8)

(Why would you want to? Sometimes it's easier!)

[57 x 5] (2) = (57) [ 5 (2) ]

(you can't hear this property! but you can SEE it!)

Aunt Sally says you must always do parentheses first, but Associative says that you can actually take the parentheses away, put parentheses in, or change where the parentheses are if you have all multiplication or all addition.


These properties give you a choice when it's all multiplication OR all addition

There are no counterexamples for these two operations.


BUT THEY DO NOT WORK FOR SUBTRACTION OR DIVISION
(lots of counterexamples!
10 - 2 does not equal 2 - 10

15 ÷ 5 does not equal 5 ÷ 15)


SO WHY SHOULD YOU CARE????

Because it makes the math easier sometimes!

Which would you rather multiply:
(2)(543)(5) OR (2)(5)(543) ???

Commutative allows you to choose!


ANOTHER EXAMPLE: [(543)(5)](2)

Aunt Sally would say you must do the 543 by the 5 first since it's in [ ]
But our friend the Associative Property allows us to simply move the [ ]

[(543)(5)](2) = (543)[(5)(2)]
which is so much easier to multiply in your head!!!



TWO MORE FRIENDS: THE IDENTITY PROPERTIES OF

ADDITION AND MULTIPLICATION
F
For addition, we know that adding zero to anything will not change the IDENTITY of what you started with: a + 0 = a
(what you started with)
0 is known as the ADDITIVE IDENTITY.


For multiplication, we know that multiplying 1 by anything will not change the IDENTITY of what you started with:
(1)(a) = a
(what you started with)
1 is known as the MULTIPLICATIVE IDENTITY.

Sometimes 1 is "incognito" (disguised!)

We use this concept all the time to get EQUIVALENT FRACTIONS.

Say we have 3/4 but we want the denominator to be 12

We multiply both the numerator and the denominator by 3 and get 9/12

We actually used the MULTIPLICATIVE IDENTITY of 1, but it was disguised as 3/3

ANYTHING OVER ITSELF = 1
(except zero because dividing by zero is UNDEFINED!)

a + b - c = 1
a + b - c


We also use this property to SIMPLIFY fractions.
We simplify all the parts on the top and the bottom that equal 1 (your parents would say that we are reducing the fraction)


6abc/2a  =  3bc  
2a/2a = 1 and that's why we can divide the fraction by it.
                    

10a/5 =     2a                                                                                              
AGAIN, WE LOVE THESE PROPERTIES BECAUSE THEY MAKE OUR LIFE EASIER!
AUNT SALLY HATES THEM BECAUSE THEY ALLOW US TO BREAK HER RULES!!!

Algebra Period 4

ABSOLUTE VALUE EQUATIONS:Section 3-8

Review of SOLVING equations:

1. Clear both sides of any decimals or fractions

2. Distribute if necessary (or divide both sides if compatible number)

3. Collect like terms on the SAME SIDE of the equation

4. Use the additive inverse to move any variables on BOTH SIDES of the equation

5. Use the addition property of equality and
then the multiplication property of equality to isolate the variable.


Generally, you solve these the same way you solve regular equations.

Make sure you balance equally on both sides!


Follow the steps of a 2 step equation.

1. Add the opposite (you can subtract as well)

2. Multiply by the reciprocal (you can divide as well)


THE DIFFERENCE?
YOU HAVE 2 POSSIBLE ANSWERS! (+ and -)


EXAMPLE: 2 l x l + 1 = 15

2 l x l + 1 - 1 = 15 - 1

2 l x l = 14

1/2 ( 2 l x l ) = 1/2 (14)

l x l = 7

x = {-7, 7}



Remember: If you get that the absolute value is negative, the answer is the NULL SET-- just like before!!

EXAMPLE: 2 l x l + 16 = 15

2 l x l + 1 - 1 = 15 - 16

2 l x l = -1

1/2 ( 2 l x l ) = 1/2 (-1)

l x l = -1/2

NOT POSSIBLE! So the answer is the null set