Transcript In the next slides you will review

Algebra 1 Review
Casey Andreski
Bryce Lein
In the next slides you will review:
Solving 1st power equations in one variable
A. Don't forget special cases where variables cancel
to get {all reals} or
B.
Equations containing fractional coefficients
C. Equations with variables in the denominator –
remember to throw out answers that cause division by
zero
Special cases
Cancel variables
3x+2=3(x-1) distribute
3x+2=3x-3 subtract 3x
2=-3
finished
Fractional Coefficient
• 1/2x - 3 + 1/3x = 2 multiply by a common
denominator
• 3x - 18 + 2x = 12 add like terms
• 5x = 40
divide by 5
• X=8
finished
Variables in the denominator
• 5/x + 3/4 = 1/2 Multiply by a common
denominator
• 5 + 3/4x = 1/2x group like terms
• 5 = -3/4x + 2/4x add like terms
• 5 = -1/4x
multiply by common
denominator
• -20 = x
Properties
Addition Property (of Equality)
Example:
a+c=b+c
Multiplication Property (of Equality)
Example:
If a = b then a x c = b x c.
Reflexive Property (of Equality)
Example:
a=a
Symmetric Property (of Equality)
Example:
a = b then b = a
Transitive Property (of Equality)
Example:
If a = b and b = c, then a = c
Associative Property of Addition
Example:
a + (b + c) = (a + b) + c
Associative Property of
Multiplication
Example:
a x (b x c) = (a x b) x c
Commutative Property of
Addition
Example:
a+b=b+a
Commutative Property of
Multiplication
Example:
axb=bxa
Distributive Property (of
Multiplication over Addition
Example:
a x (b + c) = a x b + a x c
Prop of Opposites or Inverse
Property of Addition
Example:
a + (-a) = 0
Prop of Reciprocals or Inverse
Prop. of Multiplication
Example:
(b)1/b=1
Identity Property of Addition
Example:
y+0=y
Identity Property of Multiplication
Example:
b x 1= b
Multiplicative Property of Zero
Example:
ax0=0
Closure Property of Addition
Example:
2+5=7
Closure Property of Multiplication
Example:
4 x 5 = 20
Product of Powers Property
Example:
42 x 44 = 46
Power of a Product Property
Example:
(2b)3 = 23 x b3 = 8b3
Quotient of Powers Property
Example:
54/53 = 625/125 or 54-3 = 51 = 5
Power of a Quotient Property
Example:
(4/2)2 = 42/22 = 4
Zero Power Property
Example:
a0 = 1
Negative Power Property
Example:
a-6 = 1/a6
Zero Product Property
Example:
If ab = 0 , then either a = 0 or b = 0.
Product of Roots Property
a b  a  b
Quotient of Roots Property
a
a

b
b
Root of a Power Property
Example:
x x
2
Power of a Root Property
Example:
 x
2
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. 14 + 3 = 3 + 14
Answer:
Commutative Property (of Addition)
17 = 17
In the next slides you will review:
Solving 1st power inequalities in one
variable. (Don't forget the special
cases of {all reals} and )
A. With only one inequality sign
B. Conjunction
C. Disjunction
With only one inequality sign
3+x<3+2
Click when ready to see the answerer
X<2
2
Conjunction
3+5<1+x>-2-1
Click when you’re ready to see the answer.
8<1+x>-2-1
7<x>-4
-4
7
Disjunction
3x>(14+4)
or
Click to see the answer
3x>18
x<3-4
or
x<-1
X>6
-1
6
In the next slides you will review:
Linear equations in two variables
Lots to cover here: slopes of all
types of lines; equations of all types
of lines, standard/general form, pointslope form, how to graph, how to find
intercepts, how and when to use the
point-slope formula, etc. Remember
you can make lovely graphs in
Geometer's Sketchpad and copy and
paste them into PPT.
Slope
Finding the slope with 2 given points
m = Slope
Example:
Click for an
example
(9,-3) (6,2)
2-9
-7
6+3
9
Equations of Lines
Slope intercept form- Y = Mx + B
Standard form – Ax + By = C
Point slope form- Y – Y1 = M (X – X1)
Graphing Lines
Point Slope- use this when you only have 2
points.
First : find the slope
Next put the equation into point slope form:
y-y1=m(x-x1)
Example: (3,5) (2,1)
5 1
Slope: 3  2 = 4
Y-5=4(x-3) = y-5=4x-12 = y=4x-7
Graphing Lines
Slope intercept - y=-3x+7
7= y intercept
-3 = slope
Graphing Lines
Standard form - 3x + 2y = 6
Set x to zero to find y
Set y to zero to find x
Points : (2,0) (0,3)
In the next slides you will review:
Linear Systems
A. Substitution Method
B. Addition/Subtraction
Method (Elimination )
C. Check for understanding of
the terms dependent,
inconsistent and consistent
Substitution Method
4x-5y=12
Y=2x-8
Put (2x-8) in for y for the top equation
Click for solution
4x-5(2x-8)=12 Distribute
4x-10x+40=12 add/subtract common terms
-6x=28
Divide
X= -3/14
Addition/Subtraction
Method (Elimination )
3x+5y=7
2x-4y=5
Multiply both equations to get either x or y to cancel
2(3x+5y)=7
= 6x+10y=14
Subtract
3(2x-4y)=5
=
6x-12y=15
22y=-1
Divide by 22
y= -1/22
Terms
Dependent- both same line (Infinite solutions)
Inconsistent- parallel lines (No solutions)
Consistent- Intersecting lines (One solution)
In the next slides you will review:
Factoring – since we just
completed the Inspiration Project
on this topic, just summarize all
the factoring methods quickly.
Note that you will be using your
factoring methods in areas 7 & 8
below so no need to include
extra practice problems here.
Factoring Binomials
49x4-9y2
(7x2+3y) (7x2-3y)
sum and diff of squares a3-27
(a-3) (a2+3a+9)
click for answers
difference of squares
Factoring Trinomials
GCF
Reverse foil
PST
Click for answers
2b+4b2+8b
2b(1+2b+4)
x2+5x+6
(x+3) (x+2)
4x2-20x+25
(2x-5)2
4 or More
Click for answers
3 by [(x1
2 by 2
x2+8x+16-3y2
(x+4)2-3y2
[(x+4)-3y] +4)-3y]
c3+bc+2c2+2b
c2(c+2)+b(c+2)
(c2+b) (c+2)
In the next slides you will review:
Rational expressions – try to use
all your factoring methods
somewhere in these practice
problems
A. Simplify by factor and cancel
B. Addition and subtraction of
rational expressions
C. Multiplication and division of
rational expressions
Factor and Cancel
x4
2
x  16
=
1
x4
Addition and subtraction of
rational expressions
2x
x

2
x  16 x  4
Click to see steps
2 x  x ( x  4)
2
x  16
2x  x  4x
2
x  16
2
x  6x
2
x  16
2
Multiplication and division of
rational expressions
x  4x  4 x  3


2
x  x6 x6
2
Click to see answer
( x  2)( x  2) x  3


( x  2)( x  3) x  6
x2
Division is multiplication of the
reciprocal
x6
In the next slides you will review:
Functions
A. What does f(x) mean? Are all
relations function?
B. Find the domain and range of a
function.
C. Given two ordered pairs of data,
find a linear function that contains those
points.
D. Quadratic functions – explain
everything we know about how to graph a
parabola
Functions
f(x) means that f is a function of x
All functions are relations but not all
relations are functions
A function is 1 to 1 which means for each
input there is exactly one output
Functions
Domain- Set of inputs
Range- Set of outputs
f(x)=2x-1
Domain – all real numbers
Range – all real numbers
Functions
(1,1) and (0,-1)
Are two ordered pairs of the
linear function f(x)=2x-1
Quadratic functions
f(x)=ax2+bx+c
b
Vertex x= 2a , then solve for f(x)
X-intercepts set f(x) equal to zero factor
and solve for x
y-intercepts Set x to zero and solve for f(x)
b
line of symmetry 2a the line of 2ab
In the next slides you will review:
Simplifying expressions with
exponents – try to use all the power
properties and don't forget zero and
negative powers.
Exponents
Property #1
Property #5
x0 = 1
(x × y)n = xn × yn
Example: 40 = 1 and
(2500000000000000000000)0 = 1
(6 × 7)5 = 65 × 7
Property #2
Property #6
xn × xm = xn + m
x-n = 1 ÷(xn) = 1/(xn)
Example: 46 × 45 = 46 + 5 = 411
8-4 = 1 ÷ (84) = 1 / (84)
Property #3
Property #7
xn ÷ xm = xn − m
(x/y)n = xn / yn
Example: 46 ÷ 45 = 46 − 5 = 41
(8/5)4 = 84 / 54
Property #4
Property #8
(xn)m
=
xn × m
(52)4 = 52 × 4 = 58
www.basic-mathematics.com
In the next slides you will review:
Simplifying expressions with
radicals – try to use all the root
powers and don't forget
rationalizing denominators
Expressions with Radicals
2  8  16  4
2
1
1


8
4 2
2  8  2 2 2 3 2
1
1 2
2


2
2
2 2
In the next slides you will review:
Minimum of four word problems of
various types. You can mix these in
among the topics above or put them
all together in one section. (Think
what types you expect to see on
your final exam.)
Word Problem
You drove 180 miles at a constant rate and it took
you t hours. If you would have driven 15 mph
faster you would have saved an hour. What was
your rate?
180 = rt → t = 180/r
180 = (r +15)(t –1)→180= (r+15)(180/r – 1)
180r = (r+15)(180 – r)→180r=180r-r2+2700-15r
r2+15r-2700=0→(r-45)(r+60)=0
r=45 your rate was 45 mph
Word Problem
If Joe can shovel his driveway in 2 hours and Bill can do it
in 3 hours, how long will it take for both of them to shovel
the driveway.
x x
 1
2 3
3x  2 x  6
5x  6
1
x 1
5
Word Problem
If 2 t-shirts and 3 pairs of shorts cost $69, and 2
pair of shorts are $30. How Much is a t-shirt?
2t+3s=69
2s=30
s=15
2t+3(15)=69
2t+45=69
2t=24
t=12
Word Problem
After bill lost his cell phone he had to pay his
parents 28% of the cost to buy a new
phone. Bill had to pay $21.28. What was
the price of the phone
.28  p  21.28
21.28
p
.28
p  76
$76
In the next slides you will review:
Line of Best Fit or Regression Line
A. When do you use this?
B. How does your calculator
help?
C. Give a set of sample data in
question format to see if your
students can find the regression
equation.
Line of best fit or regression
You use to come up with a linear equation that best fits the
data.
Put the input in list 1 and the out put in list 2
Then hit stat calc
Next hit 4:linreg(ax=b)
Y=ax+b is the line of best fit for the data
Question
What is the line of best fit for the given data points?
(0,5) (1,9) (-1,4) (-3,0) (-2,1) (3,13)
Y=1.5x+4.8