Download Addition Property (of Equality)

Thinking Mathematically
By: Mac Sinsky and Matt
Hipp
Solving 1st power equations in one
variable
A. Special cases where variables cancel to get all reals
Example: 2x+6=2(x+3)
-Distribute the 2
2x+6=2x+6
Cancel out to equal no set
B.
Equations containing fractional coefficients
Example: x+ 10
=5
5
10
5
=2, so x=3
Equations with variables in the denominator
Example: 28  7
x
28 divided by 7 =4, so x=4
In the next slides you will review:
All the Properties, and
then take a Quiz on identifying
the Property Names
Addition Property (of Equality)
Definition: If one number is added to two sides of an
equation
Example: If 3=3. then 2 + 3=3 + 2
Multiplication Property (of Equality)
Definition: If one number is multiplied to two sides of
an equation
Example: If 2=2, then 2 x 5=5 x 2
Reflexive Property (of Equality)
Definition: One number equals the same number
Example: 10=10
Symmetric Property (of Equality)
Definiton: One number equals the same number,
even if it’s in a different order
Example: If 6=8, then 8=6
Transitive Property (of Equality)
Definition: One number equals another number, and
that number equals another number
Example: If 12=15, and 16=12. then 15=16
Associative Property of Addition
Definition: When three numbers are added, and
changing the order has the same end result
Example: (5 + 6)+ 4=5+(4 + 6)
Associative Property of Multiplication
Definition: When three numbers are multiplied, and
changing the order has the same end result
Example: (4 x 7)x 2=4 x(2 x 7)
Commutative Property of Addition
Definition: When two numbers are added, and, if the
order is changed, the result is the same
Example: 7 + 6=6 + 7
Commutative Property of Multiplication
Definition: When two numbers are multiplied, and, if
the order is changed, the result is the same
Example: 9 x 8=8 x 9
Distributive Property (of
Multiplication over Addition)
Definition: When the distributive property equals the
distributive property broken down
Example: 4x(5+8)=4x5 +4 x 8
Prop of Opposites or Inverse
Property of Addition
Definition: When a number added to its opposite
equals zero
Example: 7 +(-7)=0
Prop of Reciprocals or Inverse Prop. of
Multiplication
Definition: when two numbers are multiplied, and the
answer is 1
Example: 4 x ¼=1
Identity Property of Addition
Definition: When 0 plus a number equals the number
that was added to 0
Example: 0 + 5=5
Identity Property of Multiplication
Definition: When 1 times a number equals the
number that was multiplied by the 1
Example: 1 x 4=4
Multiplicative Property of Zero
Definition: When 0 times a number equals 0
Example: 0 x 3=0
Closure Property of Addition
Definition: When two numbers are added to equal a
different number
Example: 5 + 3=8
Closure Property of Multiplication
Definition: When two numbers are multiplied to equal
a different number
Example: 4 x 6=24
Product of Powers Property
Definition: When two powers and their exponents are added
Example: 22 x 25=4 x 32=128 is the same as 22 + 5 =27=128
Power of a Product Property
Definition: When you want to find the power of a product
Example: (3 × 4)2 = 122 = 144 is the same as 32 × 42 = 9 ×
16 = 144
Power of a Power Property
Definition: When you want to find the power of a power
Example: (22)3 = 43 = 64 is the same as 22×3 = 26 = 64
Quotient of Powers Property
Definition: When dividing two powers that have the
same base, subtract the exponents
Example: 6 4 = 642
62
Power of a Quotient Property
Definition: When dividing two bases with the same
exponents, cancel out the common factors
Example: ( 2 )2  22  4 is the same as 22  4  4
2
22
4
Zero Power Property
Definition:When zero is the exponent the answer is 1
Example: 80  1
Negative Power Property
Definition: When the power is a negative, the answer
is 1 over that power. Moving the power to the
denominator takes away the negative.
Example: 94  1
4
Zero Product Property
Definition: When two numbers equal zero, one of the
numbers must be equal to zero
Example: If ab=0, then a=0 or b=0
Product of Roots Property
Definition: When two numbers, both inside the
root, equal both numbers in separate roots
Example:
(5)(6)
=
5x 6
Quotient of Roots Property
Definition: When a fraction is in a root, simplify the
numbers.
8 1

Example:
16
2
Root of a Power
Definition: When an exponent is in a square root
Example:
3
2
-The power and the root cancel, so final answer is 2
Power of a root
Definition: When the square root is powered
2
Example:
5
- Multiply with the power and the number, then use the root
25  5
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. 30 + 2 = 2 + 30
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 65=54, and 25=65, then 25=54
Answer:
Transitive Property (of Equality)
Solving 1st power inequalities in
one variable
A. With only one inequality sign
Example:2+x > 15
16>15, so 2+14>15
B. Conjunction
Example:3 < x+15 < 20
18 is greater than 3, but less than 20, so 3
< 3+15 < 20
C. Disjunction
Example:x+2 > 40
45 > 40, so 43+2 > 40
Linear Equations in
Two Variables
y 2  y1
x 2  x1
slopes of all types of lines:
equations of all types of lines
Example: y=3x+5
standard/general form: Ax+By=C
point-slope form: (y-y1)=m(x-x1)
how to graph: Use slope-intercept form
(y=mx+b)
Example: y=3x+5
(5,0) is y-intercept, and 3 is slope
up (3,1) from y-intercept
Linear Equations in Two Variables (cont.)
how to find intercepts:
find intercepts by using slope-intercept form: y=mx+b
how and when to use the point-slope formula:
The point-slope formula=
y  y1  m( x  x1 )
Linear Systems
Substitution Method
Definition: When you subtract one side of an equation
with another to get an answer
Example: 5x+y=10, x+4y=10
--Get y alone, solve for x
1. y=10-5x, x+4(10-5x)=10
2. x+40-20x=10, -19x=10-40
3. y=10-5(30)
y=-140
Linear Systems (cont.)
Addition/Subtraction
Method (Elimination)
Definition: When you add two equations to find
the variable.
Example: 5x+4y=4, 4x-4y=5
-Add these two equations
1. 9x=9 or x=1
-Plug in x to first equation to find y
2. 5(1)+4y=4, 4y=-1
Y=-1/4
Linear Systems (cont.)
Check for understanding of the terms dependent, inconsistent and
consistent
Dependent: When two lines are on top of each other
Example:
Consistent: When two lines cross once
Example:
Linear Systems (cont.)
Inconsistent: When two lines are parallel
Example:
Factoring
Sum and Difference of Cubes: 27 x3  125 y 3
Break the factors down: (3x  5x)(9 x2 15xy  25 y 2 )
Difference of Squares: 81a 2  25b 2
Break the factors down:
(9a)2-(5b)2
(9a-5b)(9a+5b)
Grouping 3x1: (x2+10x+25)-y2
Break the factors down: (x+5)+y(x+5)-y
GCF: 16x5-20x4+8x3
Find GCF: 4x 3
Solve: 4x3(4x2-5x+2)
Factoring (cont.)
Reverse FOIL
Example: (x+5)(5+x)
2
Solve: 5 x  25  x  5 x
PST
-PST stands for perfect square
trinomial
2
Example: (2 x  9s)
-Multiply using the power
- Multiply the number 9 times the
number of the power(2)
Solve: 4r 2 +36rs+81s
Grouping 2x2
Example: (a-9)
2
-Multiply with the
power(2)
-Multiply 9 with the
number of the power(2)
Solve:
a 2 -18a+81
Rational Expressions
A. Simplify by factor and cancel
3a  6
Example: 3a  3b
-factor, then cancel out common
factors 33aa36b  3(3(aa  b2))  a  b
Rational Expressions (cont.)
Addition and subtraction of rational expressions
Example: 3c  5c
16 16
-Add numerator, and cancel
3c  5c 8c c


16
16 2
Rational Expressions (cont.)
Multiplication of rational expressions
Example: 4 x 21
7 8
-Multiply, and simplify
4 21 84 3
x 

7 8 42 2
Quadratic equations in one variable
Factoring
Example for any terms: 5 x 2  15 x
x=3
Example for binomials: x 2  25
X=5
Example for trinomials: x 2  8 x  16
-Put zero on one side, then find PST
x 2  8 x  16  0
PST=-4
Quadratic equations in one variable (cont.)
Square root of both sides
Example: x 2  64
X=8
Complete the square
Example: 3x 2  6  x 2  12 x
- Get zero on its own side, then simplify
2 x 2  12 x  6  0
x 2  6 x  9  12
( x  3)( x  3)
x  3  12
x  3 2 3
Functions
A. What does f(x) mean?: f(x)=y
B. Find the domain and range of a function
Domain: to see how much of x is covered
Range: to see how much of y is covered
Example:
(1,1)(4,4)(5,3)
Domain is 4- From 1 to 5
Range is 3- From 1 to 4
Functions (cont.)
Given two ordered pairs of data, find a linear function that contains those
points
Example: (3,5)(7,10)
-Find the slope
10  5 5

73 4
-Put slope in point-slope formula
y 5 
5
( x  3)
4
5
15
x
4
4
5
35
y  x
4
4
y 5 
Functions (cont.)
Quadratic functions
Formula: y  ax 2  bx  c
Example: y  x 2  2 x  8
Because a=1, graph opens up
To find x-intercept, set y to 0
0=(x-4)(x+2)=(4,0)(-2,0)
To find y-intercept, set y to 0
f (0)  (0)2  2(0)  8  (0, 8)
Vertex: formula=
2

 (1, 9)
2(1)
b

2a
Axis of symmetry: x=1
The discriminant ( b2  4ac ) tells you the number of x-intercepts.
Simplifying Expressions
with Exponents
When two exponents are multiplied, add
2
2
4
(
x
)(
x
)

x
Example:
When an exponent is multiplied by a power, multiply the
powers
Example: ( x 2 )3  x 6
If an exponent has a 0 power, the answer is 1
Example: x 0  1
When an exponent has a negative power, the answer is a
fraction
1
2
Example: x  2
x
Simplifying Expressions
with Radicals
When square root has a second power,
the square root and power cancel out
2
Example: x  2  (15)2
x+2=225
x=223
Word Problems
Mike spent $95 on Packer tickets. This was $25 less than
twice of what John spent. How much were John’s
tickets?
2x-25=95
2x=70
X=35
John’s Packer tickets cost $35.
Word Problems (cont.)
There are n number of players on a basketball team. This is four less
than two times the amount of coaches. How many coaches are on
the team?
2x-4=n
2x=b+4
X=2
There are two coaches.
Word Problems (cont.)
 The sum of two consecutive numbers is 21. What are the
numbers?
2x=21-1
2x=20
X=10
The numbers are 10 and 11
Word Problems (cont.)
Two students are running for class president. One student
got 30 more votes than the other student. If the total
amount of votes is 45, how much votes did each person
get?
x+x+30=35
One x is bigger than the other x, so the first x is 10, and the
second x is 5
Line of Best Fit or
Regression Line
When do you use this?:
Linear regression on the TI-84 calculator is used
when you want to find the line of best fit, or when you
want to see the stat plots. You can also see both of
these together.
How does your calculator help?:
Your TI-84 calculator can find out and graph your line
of best fit
Example: (-18,4)(-11,7)(-2,12.5)(1,14)(6,16)