Algebra Honors (Period 6 & 7)

Adding & Subtracting Polynomials 4-2

Terms to Know:
Polynomials = SUM of monomials

Monomials must have variables with whole number powers.
no variables in the denominator, no roots of numbers!!
so 1/x is not a monomial
neither is x ^1/2

constants have whole number power of zero..
7 is really 7x⁰
constants are also called constant monomials


1 term = monomial
2 terms = binomial
3 terms = trinomial

TERMS are separated by addition
( if see subtraction-- THINK: add the opposite!!)

Coefficient - number attached to the variable ( it can be a fraction)
3x² - 10x
the coefficients are 3 and -10. Make sure to attach the negative sign to the coefficient -- and ADD the OPPOSITE

y/6 is really (1/6)y so the coefficient is 1/6
if you have -x/3 that is really (-1/3)x so the coefficient is -1/3


Constant = the number that is not attached to ANY variable

Two monomials that are exactly alike ( except for the coefficient) are said to be similar, or like, terms.
-5xy² and xy² and (1/3) xy² are like terms. So is 16yxy because when you combine it 16yxy becomes 16xy²
But... -3xy² and -3x²y are NOT!!!

A polynomial is simplified when no two terms are similar
-6x³ + 3 x² + x² + 6x³ - 5 can be simplified to
4x² - 5



Some MORE TERMS YOU NEED TO KNOW
Degree of a variable in a term = number of times that variable occurs as a factor.
Degree of a term = SUM of the exponents of all its variables
-6x⁴ : the degree is 4
8x² : the degree is 2
-2x : the degree is 1
9 : the degree is 0 ( think 9 is really 9x⁰

Degree of a polynomial - HIGHEST degree of any of its terms
so
-6x⁴ + 8x² + -2x + 9
The degree of the polynomial is : 4

Leading term = term with the HIGHEST degree
Leading coefficient- the coefficient of the leading term






Descending order- write the variables with the highest power first ( This is the way it is usually written)

Ascending order- write the variables with the lowest power first ( actually NEVER used in practice)


Adding Polynomials
This is nothing more than combining LIKE TERMS
LIKE TERMS = same variable AND same power

You can either do this using 3 different strategies:
1. Simply do it in your head, but keep track by crossing out the terms as you use them.
2. Rewrite putting the like terms together (commutative and associative property)
3. Rewrite in COLUMN form, putting like terms on top of each other like you do when adding a column of numbers.

EXAMPLE OF COLUMN FORM:
(5x⁴ - 3x² - (-4x) + 3) + (-10x⁴ + 3x³- 3x² - x + 3)
Rewrite in column form, lining up like terms:


Subtraction of Polynomials
You can use the ADDITIVE INVERSE PROPERTY with polynomials!
Subtracting is simply adding the opposite so.............
DISTRIBUTE THE NEGATIVE SIGN TO EACH TERM!!
(Change all the signs of the second polynomial!)
After you change all the signs, use one of your ADDING POLYNOMIAL strategies!
(see the 3 strategies listed above under Chapter 5-7)

EXAMPLE OF COLUMN FORM:
(5x⁴ - 3x² - (-4x) + 3) - (-10x⁴ + 3x³- 3x² - x + 3)
Rewrite in column form, lining up like terms:
5x⁴ - 3x² - (-4x) + 3
- ( -10x⁴ + 3x³- 3x² - x + 3)
-----------------------------------

For the sake of showing you here, I have added ZERO Terms to line up columns
+ 5x⁴ + 0x³ - 3x² -(-4x) + 3
-(-10x⁴ +3x³- 3x² - x + 3)
-----------------------------------

DISTRIBUTE THE NEGATIVE, THEN ADD:
5x⁴ + 0x³ - 3x² - (-4x) + 3
+10x⁴ -3x³ +3x² + x - 3
-----------------------------------
15x⁴ - 3x³ + 5 x

Algebra Honors (Period 6 & 7)

Exponents 4-1
In the expression 5⁴ the number 4 is called the exponent and the number 5 is called the base
We call 5⁴ the exponential form of 5⋅5⋅5⋅5 (which is the expanded form)
The exponent tells you the number of times the base is used as a factor.
bⁿ = b⋅b⋅b⋅b⋅b⋅⋅⋅⋅b ( a total of n factors)
The expression bⁿ tells you that b is used as a factor n times.

-2⋅p⋅q⋅3⋅p⋅q⋅p written in exponential form is -6p³q²

Be careful when an expression contains both parentheses and exponents
(2y)³ = (2y)(2y)(2y) = 8y³
and
2y³ means 2⋅y⋅y⋅y = 2y³
-3⁴ = -81
(-3)⁴ = (-3)(-3)(-3)(-3) = +81
(1 + 5)² = 36
1 + 5²= 26
Simplify
(x –y)³/2x + y
for x = 2 and y = 5

-3

Simplify
x⁴ – a⁴
for x = 2 and a = 3
-65

[2³ + 3³] ÷ [2³ + (-1)²] = (8 + 27) /9 = 35/9


Evaluating terms - this is what we have been doing all year... plug it in, plug it in!!
Remember to ALWAYS put the number you substitute in parentheses!!

2x²y + 5xy - 4, where x = -4 and y = 5
Substitute carefully:

2(-4)²(5) + 5(-4)(5) - 4
= 2(16)(5) +(-20)(5) - 4
= 160 +(-100) -4
= 160 -104
= 56

Math 6 Honors ( Periods 1, 2, & 3)

Writing Inequalities 2-3

2 < 7 and 7 > 2 are two inequalities that state the relationship between the numbers 2 and 7

2 < 7 reads 2 is the less than 7 and 7 > 2 reads 7 is greater than 2
The symbols < and > are called inequality symbols.

Notice the mathematical sentence (inequality)
Two is less than ten or 2 < 10 is different from the mathematical phrase (expression) Two less than ten. 10 - 2 A number 2 + x is greater than a number t 2 + x > t

The point of the number line that is paired with a number is called the graph of that number.

Check out the graph in the middle of page 39 of our textbook. When you graph numbers on the number line, make sure to place a dot DIRECTLY ON the number line at that particular number's location.
Again, check out our textbook for examples!!

Looking at the graph of numbers, we see that the larger number will be to the right of the smaller number.

A number n is between 6 and 12 so 6 < n < 12 or 12 > n > 6

Thursday's Lesson: continuing on ....

Notice the subtle differences in the sentence
Six is greater than a number t
and the phrase
six greater than a number t

Six is greater than a number t becomes 6 > t
while
six greater than a number t becomes t + 6

What about the following inequality:
A number p is greater than a number q
is p > q

The value in cents of d dimes is less than the value in cents of n nickels.

If you need to-- set up your T-charts (refer to your class notes) one for dimes and the other for nickels.

10d represents the number of dimes and 5n represents the number of nickels

so 10d < 5n

Math 6 Honors (Period 6 and 7)

Writing Mathematical Equations 2-2

The process of writing equations really is just writing two equal expressions and joining them by an equals sign. The words "is" "equals" or "equal to" all indicate that two phrases NAME the same number.
The equals sign is the VERB in a mathematical sentence-- without it you have a mathematical expression!!
Eight increased by a number x is equal to thirty-seven.
In translating this mathematical sentence, I always start but placing the equals sign directly under the words "is equal"
so my first step would be

Eight increased by a number x is equal to thirty-seven.
---------------------------------> = <------------------
Then I would translate each mathematical phrase separately.
Yes, thirty-seven is a mathematical phrase!!
8 + x = 37

Ten is two less than a number n
10 = n - 2

Twice a number w equals the sum of the number and four
2w = n + 4

Notice that when you are indicating multiplication the number (coefficient) ALWAYS is placed in front of the variable.

So three times a number b would be 3b
The only time you see the letter first-- is when you are looking for our ROOM--
which is K 101... a mathematician did not label the room numbers!! :)

Sometimes we need to write an equation for a word sentence that involves measurements. MAKE SURE that each side of the equation uses the SAME UNIT of Measurement!!
For example. Write an equation for: The value of d dimes is $27.50
WE know that the value of d dimes is 10d cents (from our previous lesson) .. but $27.50 is in terms of dollars so we need to change it to cents . $27.50 is 2750 cents ... So our equation becomes 10d = 2750.

Algebra Honors (Period 6 & 7)

Proof in Algebra 3-8
Some of the properties discussed in the previous chapters are statements we assume to be true. Others are called theorems. A theorem is a statement that is shown to be true using a logically developed argument. Logical reasoning that uses given facts, definitions, properties, and other already proven theorems to show that a particular theorem is true is called a proof. Proofs are used extensively in Algebra as well as Geometry.
Prove: For all numbers a and b, (a + b) – b = a

Many times, only the KEY reasons are states—the substitute principle and the properties of equality are usually not stated. So, the above prove could be shortened to 4 steps:

Prove: For all real numbers a and b, such that a≠ 0 and b ≠0
1/ab = 1/a ⋅1/b

Since 1/ab is the unique reciprocal of ab, you can prove that

1/ab = 1/a⋅ 1/b by showing that the product of ab and 1/a ⋅ 1/b is 1

Once a theorem has been proved, you can use it as a reason in other proofs. Check the Chapter summary on page 88 of our textbook for the listing of properties and theorems that you can use as reasons in your proofs for our homework.

Algebra Honors (Period 6 & 7)

Cost Income, and Value Problems 3-7
Objective: To organize the facts of a problem in a chart & solve problems involving cost, income, and value
Using a chart to organize the facts of a problem can be a helpful problem solving strategy.
Cost = number of items X price per item
Income = hours worked X wage per hour
Total value = # of items X value per item

Example: Tickets for the senior class play cost $6 for adults and $3 for students. A total of 846 tickets worth $3846 were sold. How many student tickets were sold?
Let x = the number of student tickets sold
Then 846- x = the number of adult tickets sold

The only fact NOT recorded in this chart is that the total cost of the tickets is $3846.
The equation becomes
3x + 6(846 –x) = 3846.
3x + 5076 – 6x = 3826
-3x = -1230
x = 410
Check to make sure what x represented… in this case the number of student tickets so
410 student tickets were sold.


We then turned to Problem 4 on Page 128

A collection of 52 dimes and nickels is worth $4.50. How many nickels are there?
Let d = the number of dimes, so If there are 52 in the collection
52-d must = the number of nickels. So our Chart looks like:

This time we reread the given facts and with the second fact, we realize that we can add the two expressions and set them equal to $4.50
10d + 5(52 - d) = 450
solving this equation, we find that d = 38
Make sure to reread the question… it asked “how many nickels?”
So we need to use 52-d and substitute in 38… 52 -38 = 14
14 nickels is the correct answer


Next we complete # 6
Celia bought 12 apples, ate two and sold the rest at 20 cents more per apple than she paid. Her total profit was $1.00 How much did she sell each apple for?
Let b = the price she bought each of the apples for
We glued in the “yellow-colored” chart here and completed it to look like:

Now, her profit is the difference between what she sold them for and what she bought them for… Therefore the equation becomes
10(b + 20) – 12b = 100
10b + 200 -12b = 100
-2b = -100
b = 50
Re reading the question we realize we solved for what she bought the apples each for. We need to add 20cents to find out what she sold them for

She sold each apple for 70 cents.

The last problem we did from our textbook was # 14 on Page 129

Jo has 37 coins ( nickels, dimes & quarters) for $5.50 She has 4 more quarters than nickels. How many dimes does Jo have?
Let n = the number of nickels
n + 4 = the number of quarters
So if there was a total of 37 the rest must be dimes
37 – [ n + (n+4)] or 37 –(2n +4)
Let’s set up the chart

Now we know that all of them combined equal $5.50
so
5n + 10[37-(2n+4) + 25(n +4) = 550
5n + 370 – 20n – 40 + 25n + 100 = 550
combining all the n’s

10n + 330 +100 = 550
10n = 120
n = 12
Re reading the question, we find we need to see how many dimes so substitute in
37 -[2(12) +4] = 37 – [24 +4] = 37 -28 = 9
Jo had 9 dimes.

Algebra Honors (Period 6 & 7)

Problem Solving: Using Charts 3-6
Objective: To organize the facts of a problem in a chart.
Using a chart to organize the facts of a problem can be a helpful problem solving strategy.

Organize the given information in a chart
A swimming pool 25 m long is 13 m narrower than a pool 50 m long
There are two different charts you could create:

In the first chart we started with
Let w = the width of the 2nd pool; so to write the 1st pool in terms of the second we would place w-13 in place of the ?

In the 2nd chart we started with
Lt w = the width of the 1st pool; so we write the 2nd pool in terms of the first, so we would place w + 13 in place of that ?


We then solved the following:

Find the number of calories in an apple & a pear…

1) the pear contains 30 calories more than the apple.
2) Ten apples have as many calories as 7 pears.
Let a = the number of calories in an apple
Then a + 30 = the number of calories in a pear.
We glued in the “orange- colored” chart here and completed it to look like:

This time we reread the given facts and with the second fact, we realize that we can set
The last column expressions equal to each other
10a = 7(a + 30)
solving this equation, we find that a = 70
Therefore, an apple has 70 calories and a pear has 100 calories
Next we used the following two given facts to set up a chart and create an equation
1) An egg scrambled with butter & milk has 1 more gram of protein than an egg fried in butter.
2) Ten scrambled eggs have as much protein as a dozen fried eggs.
Let x = the number of protein in a fried egg.
Then x + 1 = the number of protein in a scrambled egg.
We glued in the “orange- colored” chart here and completed it to look like:

Our equation would be 10(x + 1) = 12x
We then turned to our textbook to Page 122-123 and completed problems 1 and 3 in our spiral notebook as follows:

Solve each problem using the two given facts. Complete a chart to help you solve each problem

1. Find the number of full 8 hour shifts that Maria worked last month
1) She worked twice as many 6 hour shifts as 8 hour shifts
2) She worked a total of 280 hours.

Math 6 Honors ( Periods 1, 2, & 3)

Writing Mathematical Expressions 2-1

Make sure to glue the 'purple 1/2 sheet' of math word phrases that we associate with each of the four basic operations -- into your spiral notebook (SN)

We can use the same mathematical expression to translate many different word phrases
Five less than a number n
The number n decreased by five
The difference when five is subtracted from a number n

All three of those phrases can be translated into the variable expression

n-5

The quotient of a number y divided by ten becomes y/10. It may look like only a fraction to you-- but if you read y/10 as always " y divided by 10" you have used the proper math language.

Twelve more than three times a number m

Wait-- where are you starting from... in this case you are adding 12 to 3m so you must write

3m + 12


Not all word phrases translate directly into mathematical expressions. Sometimes we need to interpret a situation.. we might need to use relationships between to help create our word phrase.

In writing a variable expression for the number of hours in w workdays, if each workday consists of 8 hours...

8w would be our expression


Some everyday words we use to so relationships with numbers:
consecutive whole numbers are whole numbers that increase by 1 for example 4, 5, 6
So the next consecutive whole number after w is w + 1.

A preceding whole number is the whole number that is 1 less and the next whole number is the whole number that is 1 greater.

So the number which precedes x would be x - 1.
The next number after n is n + 1


Thursday, September 15th Lesson:


What if I asked what is the next consecutive EVEN number after the even number "m"
It would be m + 2





What would it be if I asked what was the next consecutive odd number, after the odd number x?
x + 2
Let's look at those relationships like the workdays from Tuesday's lesson... We are going to find another strategy to use for some of the more complicated expressions...

First set up a T chart- as discussed in class
put the unknown on the left side of the T chart... The unknown is always the one that reads like " w workdays"

so in this case

w workdays on the left side and under it you put
1
2
3

On the right side put the other variable-- in this case hours
under hours put the corresponding facts you know-- the relationship between workdays and hours as given in this case
hours
8
16
24

all of those would be on the right side of the T chart.

Now look at the relationships and ask yourself--
What do you do to the left side to get the right side?

and in this case

What do you do to 1 to get 8?
What do you do to 2 to get 16?
What do you do to 3 to get 24?

Do you see the pattern?

For each of those the answer is "Multiply by 8" so
what do you do to w-- The answer is Multiply b 8
so the mathematical expression in this case is "8w."

What about writing an expression for
The number of feet in i inches

i inches is the unknown... so that goes on the left side of the T chart... with feet on the right

i inches ___feet
12...............1
24...............2
36...............3

I filled in three known relationships between inches and feet Now, ask your self those questions again...
What do you do to the left side to get the right side?

and in this case

What do you do to 12 to get 1?
What do you do to 24 to get 2?
What do you do to 36 to get 3?

In each of these, the answer is divide by 12
so What do you do to i? the answer is divide by 12
i inches ___feet
12...............1
24...............2
36...............3
i................i/12



and it is written i/12

Algebra Honors (Period 6 & 7)

Equations w/ The Variable on Both Sides 3-5

Partial Notes
2 unique types of equations:
Identity equation: You solve it and you get the same thing on both sides...if you solve until you cannot do anything more, you get 0 = 0.
What this means is that you can pick any number and the equation will work!
The Distributive Property is the simplest example of an Identity Equation:
3(x + 7) = 3x + 21  
Distribute on the left side and you'll get:
3x + 21 = 3x + 21  
At this point, you should already know this is an Identity!
If you keep solving, you would subtract 3x from each side and you'll get:
21 = 21
and you know again that this is an Identity.
If you now subtract 21 from each side:
0 = 0
BUT I WOULDN'T GO THIS FAR! AS SOON AS YOU HAVE THE SAME THING ON BOTH SIDES, YOU CAN STOP AND SAY IT'S AN IDENTITY EQUATION!!!

Null set equation:
You solve it and you get an impossible answer:
3(x + 7) = 3x + 10
3x + 21 = 3x + 10
21 = 10
WHEN WILL THAT HAPPEN??? NEVER!!! SO THERE IS NO POSSIBLE SOLUTION TO THIS! The answer is the null set.

Math 6 Honors ( Periods 1, 2, & 3)

Problem Solving 1-6 & 1-7

Follow the 5-step plan... handed out in class.. make sure to glue it into your Spiral Notebook,

MAKE SURE TO READ and RE READ the problem!!

When you are finished--> check your work to make sure your answers make sense .
LABEL, LABEL, LABEL

Sometimes you need to use more than one operation.

We turned to Page 19 and looked at problems 1, 2, 4 and 5 .. If you missed class on Friday, make sure to review those problems.
Then we looked at Example 2 on Page 21

Simon had $165 in his checking account. He wrote checks for $32, $19, and$47. How much did Simon have left in his account?

When you write checks, you will subtract them from your balance. First add all the checks that you wrote 32 + 19 + 47 = 98. So that's $98 to subtract from the beginning balance of $165.
165- 98 = 67
So Simon had $67 left in his checking account.

Then we looked at Page 22 the Class Exercises and discussed what operations had to be used for problems # 2 and # 4.

Make sure to check out those problems.
#2
1st OP: addition
2nd OP: division

#4
1st OP: multiplication
2nd OP: addition

Math 6 Honors ( Periods 1, 2, & 3)

Order of Operations 1-5

O³ because Order Of Operations

Grouping symbols ( ) hugs!! so important in middle school...and in math!!
[ ] brackets and { }


These Grouping symbols show which operations are performed first.
Even the Fraction bar is a grouping symbol!!

8 + 3 - 9⋅ 2 ÷ 3

Notice there are NO grouping symbols. WE then tried to come up with a number and in several of our classes we have over 6 different answers... we can't have that happening!!

So follow these rules:
When there are NO grouping symbols
1) Do ALL of the multiplication or division in ORDER from Left to Right
2) THEN do ALL of the addition or subtraction in ORDER from left to right!!

We tried it again... with all sorts of beeps.... and discovered the funnel process

8 + 3 - 9⋅ 2 ÷ 3
8 + 3 - 18 ÷3
8 + 3 -6
11-6
5

WE then tried it on
72 -24 ÷ 3
72 - 8
64

and then

8 + 6 ⋅ 12 ÷ 4 + 6
8 + 72 ÷ 4 + 6
8 + 18 + 6
26 + 6
32

The last one we did in class was
126 - 9÷ 3 ⋅ 28
126 -3 ⋅ 28
126 - 84
42


if you have a series of grouping symbols... do the inside ones first.. called nesting

{ [ ( )]}

(14 + 77) ÷ 7
91 ÷ 7
13

Make sure to show your division as a side bar. Be proud of your work!!
[3 + (4 ×5)]×10
[3 + 20]×10
[23]10
230

P--> Please --> Parentheses ( )
E --> Excuse --> Exponents e²
M --> My ----> Multiplication
D --> Dear ---> division
A--> Aunt --> Addition
S-->Sally --> Subtraction

PEMDAS but we talked about how it should really be
P
E
MD
AS

or
P
E
DM
SA

Because it is multiplication or division, whichever comes first!!
and then addition or subtraction, whichever comes first


After we have checked grouping symbols...
do all the multiplication and division in order from left to right
then,
do all the addition and subtraction in order from left to right

97 -7 5÷ 5×3 + 68
as we scan... or used our typewriter skills...(where are your sound effects???)
we discover we must divide first
funneling down, we get
97 -15× 3 + 68
97 -45 +68
52 + 68
120

what if
t = 12
w = 10
x = 9
y = 4
z = 3

tx- wz
(12)(9) - (10(3)
108 - 30
78

x(y + w)
9(4 +10)
9(14)
126

Algebra Honors (Period 6 & 7)

Using Several Transformations 3-3

a+b - b = a
and ab ÷ b = a
Use inverse operations
We can use those to solve equations with more than one step
5n - 9 = 71
by adding 9 to both sides of the equation ( using the +prop=)
we get 5n = 80
By then dividing BOTH sides by 5 using the ÷prop=
we arrive at n = 16
Using set notation {16}

(½)x + 3 = 9
By subtracting 3 from both sides using -prop=
we get ½x = 6
By multiplying BOTH sides by the reciprocal of ½ which is 2 (using xprop=)
we get
x = 12

w-5 = 2
9

multiply BOTH sides by 9 using xprop=
we get
w-5 = 18
adding 5 to both sides using +prop=
w = 23
{23}

32 = 7a + 9a
Combine Like terms FIRST
32 = 16a
2 = a {2}

4(y +8) - 7 =15
You must use the Distributive Property first
4y + 32 - 7 = 15
combine like terms on the left side
4y + 25 = 15
subtract 25 from both sides
4y = -10
divide by 4
y = -10/4
simplify BUT leave as an improper fraction
y = -5/2
Using set notation {-5/2}


But what about
-4(m +12) = 36
Yes you could do the Distributive Property BUT... why not divide both sides by -4 FIRST
-4(m + 12)= 36/-4
-4

m + 12 = -9
subtract 12 from both sides using the -prop=
and get
m = -21
{-21}
(c + 3) -2c -(1-3c) = 2

BEFORE WE COMBINE terms let's look at
-(1-3c) = -1 + 3c that is the inverse of a sum!!

c + 3 -2c -1 + 3c = 2
combining like terms we get
2c + 2 = 2
2c = 0
c = 0
{0}

Math 6 Honors ( Periods 1, 2, & 3)

Distributive Property 1-4
Distributive Property of Multiplication with Respect to Addition
DP+
a(b +c) =ab + ac
(b+c)a = ba +ca

Distributive Property of Multiplication with Respect to Subtraction
DP-
a(b -c) = ab -ac
(b -c)a = ba -ca
What's this property all about.. it is about making it easier for you to simplify!!

13(15) you could use your 5th grade skills and multiply it out or...
13(15) = 13(10 +5) = 130 + 65 = 195
OR
(10 +3)(15) = 10(15) = 3(15) = 130 + 45 = 195
SAME SOLUTIONS
What about
23(11) hmmm...
(20+3)11 = 20(11) + 3(11) = 220 + 33 = 253
OR
23(10 +1) = 23(10) + 23(1) = 230 + 23 = 253

we then took a little sidebar and discuss the trick of elevens.... If you missed that explanation.. ask a fellow student to explain multiply 2 digit numbers by 11... You will love it!!


(101)34) = (100+1)(34) = 100(34) + 1(34)= 3400 + 34 = 3434

What about 54(29) What could you do to make that easier than just multiplying out???
54(29) = (50 +4)(29) = (50)(29) + 4(29) is that any easier? What about...
54(20 +9) = 54(20) + 54(9) ... hmmm... is that easier?? What about..
54(30-1) = 54(30) - 54(1) that looks easier...
We took the time to try all three methods and discovered that the results were all 1566!!
Thank goodness!!
9× 57 What could we do here?
9(50 +7) = 450 +63 =
OR
9(60 -3) = 540 -27
OR
(10-1)57 = 570 - 57
and we discovered again... they all equaled 513


If you have...
(7 × 9) + (13 ×9)
Look and see what they share...
They both have a nine
Pull that out and put one set of HUGS ( ) with the sign that is in the middle of the two
in this case it is +
so you have
( + )9
Now look at each term, what is the other number in each of the HUGS
7 is in the first one and 13 is in the second so
(7 +13)9 NOW in this case, you want to add them BEFORE you multiply
(7 +13)9 = (20)9= 180 THat's too easy that way!!

(117 × 4) - (17 × 4)
Ask yourself.. what do they share? 4
Build your HUGS with the sign in the middle --> in this case it's subtraction
( - )4
Now put the other terms in place
(117-17)4 = (100)4 = 400 OMG that is way too easy now!!


(8× 6) + (2 × 6)
well we could just do it.. like 5th grade and get
48 + 12 = 60
or we could start the DP+
(8+2)6 and still get 48 + 12 = 60
OR
you could use the DP+ to make it even simpler
(8 +2)6 = (10)6 = 60 in 2 seconds flat!!

Algebra Honors (Period 6 & 7)

Transforming Equations: All Four Op's 3-1 & 3-2

Addition Property of Equality
for all real numbers, a, b, and c
and given a= b
then
a + c = b + c
and c + a = c + b
we abbreviated it as
+prop=
Make sure to read this as the Addition Property of Equality
Now
a -c = b - c
is the Subtraction Property of Equality
-prop=
But realize with a tiny change
a + -c = b + -c you have the +prop=

x - 8 = 17
If we add 8 to both sides ---> That's using the +prop=
x -8 + (+8) = 17 + (+8)
x = 25


x -5 = 9
If we subtract 5 from both sides--> we are using the -prop=
x +5 (-5) = 9 + (-5)
x = 4

Multiplication Property of Equality
ac = bc
and ca = cb
and we write it
xprop=

Division Property of Equality
where c ≠ 0
a/c = b/c
and we write that
÷prop=

(-2/3)t = 8
If we use the Multiplication Property of Equality and
multiply BOTH SIDES by the reciprocal of -2/3
we have,
(-3/2)(2/3)t = 8(-3/2)
t = -12

⎮m⎮ =6
2
using the Multiplication Property of Equality (xprop=)

2⎮m⎮ =6(2)
2
⎮m⎮ =12
so m = -12 and 12
and using the set notation
we have the solution set as {-12, 12}