Download Addition Property (of Equality)

Thinking Mathematically
A review and summary of
Algebra 1
By: Bryan McCoy and Mike Pelant
In the next slides you will review:
All of the Properties and Equations
needed to succeed in the upcoming
Exam
In the next slides you will
review:
Review all the Properties and
then take a Quiz on identifying
the Property Names
Addition Property (of Equality)
Example:
If a=b then a+c=b+c
Multiplication Property (of Equality)
Example:
If a=b then a(c)=b(c)
Reflexive Property (of Equality)
Example:
If a=b then b=a
Symmetric Property (of Equality)
Example:
If a=b then b=a
(order does not matter)
Transitive Property (of Equality)
Example:
If a=b and b=c then a=c
Associative Property of Addition
Example:
(1+2)+3=6 1+(2+3)=6
(Does not matter where you put the parenthesis)
Associative Property of
Multiplication
Example:
(1  2)  3  6 1  (2  3)  6
Commutative Property of
Addition
Example:
5+3+2=3+5=2
Commutative Property of
Multiplication
Example:
24  42
Distributive Property (of
Multiplication over Addition
Example:
3(2+7-5)=3(2)+3(7)+(3)(-5)
Prop of Opposites or Inverse
Property of Addition
Example:
+8-8=0
Prop of Reciprocals or Inverse
Prop. of Multiplication
Example:
Identity Property of Addition
Example:
4+0=4
Any number plus 0 equals the original number
Identity Property of Multiplication
Example:
Any number times 1 will equal itself
4 1  4
Multiplicative Property of Zero
a0  0
Example:
A number times 0 equals 0
Closure Property of Addition
Example:
.
Closure Property of Multiplication
Example:
Product of 2 real numbers = a real
number
5  7  35
Product of Powers Property
Example:
72 × 76 = 78
Power of a Product Property
Example:
32 · 42 = 122
Power of a Power Property
Example:
Quotient of Powers Property
Example:
Power of a Quotient Property
Example:
Zero Power Property
Example:
(-3)0 = 1
Negative Power Property
Example:
Zero Product Property
Example:
if ab = 0, then either a = 0 or b
= 0 (or both).
Product of Roots Property
The product of the square
roots is the square root of the
product.
Quotient of Roots Property
For any non-negative (positive or 0) real
number a and any positive real number
b:
=√a
-√b
Root of a Power Property
Example:
Power of a Root Property
Example:
Now you will take a quiz!
Look at the sample problem and
give the name of the property
illustrated.
Click when you’re ready to see the answer.
1. a + b = b + a
Answer:
Commutative Property (of Addition)
Now you will take a quiz!
Look at the sample problem and
give the name of the property
illustrated.
Click when you’re ready to see the answer.
2. If a=b then a(c)=b(c)
Answer:
Multiplication Property (of Equality)
In the next slides you will review:
Solving inequalities
Sample Problem:
-5x < 10
x > -2
Solution Set: {x: x > -2}
2
- Remember the Multiplication
Property of Inequality! If you
multiply or divide by a
negative, you must reverse the
inequality sign.
Linear Equations in 2 Variables
Here’s a sample problem: can you graph
this: y=x-5?
Linear Systems
Can you solve this? y = 3x – 2
y = –x – 6
Now solve for Y
Y=-x-6
3x-2=-x-6
4x=8
X=2
Y=3(2)-2
Y=6-2
Y=4
The answer is (2,4)
In the next slides you will
review:
All of the Factoring
Methods
Find GCF!
Finding the GCF will make the problem
simpler greatly.
Ex:
– 2x-4y=8
– GCF = (2)
– = (2)(x-y=4)
The following is a list of the rest
of the properties:
Difference of Squares
Sum/Difference of Cubes
PST
Reverse Foil
Factor by Grouping(4 or more terms)
Rational Expressions
Try this
Problem:
Functions
f(x)= is another way to write y=
Functions are relations only when every input has a
distinct output, so not all relations are functions but all
functions are relations.
Let’s say you had the points (2,3) and (3,4) and you needed to find a
linear function that contained them. This is how you would do that.
3-4 over (divided by) 2-3 (rise over run, Y is rise, X is run)
you would get -1 over -1. This equals 1, which will be the slope. To
find y-intercept, substitute: 2=1(3)+b
2=3+b  -1=b
So your final equation is: Y=X-1. You can now graph this.
Parabolas
See if you can graph this one: x2-6x+5
The x-intercepts are (5,0) and (1,0).
y-intercept:
Vertex:
and
So the vertex is (3, -4).
Now just graph it.
Simplifying Expressions With
Exponents
Simplify this:
The "minus" on the 2 says to move the variable; the "minus" on the 6 says that
the 6 is negative. Warning: These two "minus" signs mean entirely different
things, and should not be confused. I have to move the variable; I should not
move the 6.
Your answer is:
Simplifying expressions with
radicals
Try this one:
Word problems
You need a 15% acid solution for a certain test, but your supplier only ships a 10% solution and a
30% solution. Rather than pay the hefty surcharge to have the supplier make a 15% solution, you
decide to mix 10% solution with 30% solution, to make your own 15% solution. You need 10 liters
of the 15% acid solution. How many liters of 10% solution and 30% solution should you use?
Let x stand for the number of liters of 10% solution, and let y stand for the number of liters of 30%
solution. (The labeling of variables is, in this case, very important, because "x" and "y" are not at
all suggestive of what they stand for. If we don't label, we won't be able to interpret our answer in
the end.) For mixture problems, it is often very helpful to do a grid:
Word Problems Continued
A collection of 33 coins, consisting of nickels, dimes, and quarters,
has a value of $3.30. If there are three times as many nickels as
quarters, and one-half as many dimes as nickels, how many coins of
each kind are there?
I'll start by picking and defining a variable, and then I'll use translation to convert this exercise into
mathematical expressions.
Nickels are defined in terms of quarters, and dimes are defined in terms of nickels, so I'll pick a
variable to stand for the number of quarters, and then work from there:
number of quarters: q
number of nickels: 3q
number of dimes: (½)(3q) = (3/2)q
There is a total of 33 coins, so:
q + 3q + (3/2)q = 33
4q + (3/2)q = 33
8q + 3q = 66
11q = 66
q=6
Then there are six quarters, and I can work backwards to figure out that there are 9 dimes and
18 nickels.