Solving First Power Equations in one Variable
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Transcript Solving First Power Equations in one Variable
Math Exam Review Semester 2
By Kyle Skarr and Ryan McLaughlin
Solving First Power Equations in one
Variable
Example problem 4x=24-2x
How to solve
4x=24-2x
+2x +2x
6x=24
/6 /6
X=4
Solving First Power Equations in
one Variable continued
Equations
containing
fraction
coefficients
–
Example
equation
x 3 x
5 4 2
Least common
denominator is 20
4x 15 10x
6x 15 x 2.5
Solving First Power Equations in one
Variable continued
Equations with
variables in the
denominators–
Example
10 5
x 2x
Multiply by 2x because it
is the least common
denominator
10
5
2x 2x
x
2x
20 5 25
Solving First Power Equations in one
Variable continued
Special cases– Example 5x 25 25 5x
–
Example
8x 16 8x
xx
16 0
All real
No solution
Properties
Addition Property of Equality
If a=b then a+c = b+c and c+a = c+b
Properties
Multiplication Property of Equality
If a,b,c are any real numbers and a=b then
ca=cb and ac=bc
Properties
Reflexive Property of Equality
If a is a real number then a=a
Properties
Symetric property of equality
a=b then b=a
Properties
Transitive property of equality
If a=b and b= c then a=c
Properties
Associative property of Addition
(a+b) + c = a + (b+c)
Properties
Associative property of multiplication
(ab)c = a(bc)
Properties
Commutative Property of Addition
a+b = b+a
ab=ba
Properties
Commutative property of multiplication
45 5 4
Properties
Distributive Property
a(b+c) = ab+ac
Properties
Prop. Of opposites or inverse property of
addition
5+(-5)=0
Properties
Property of reciprocals or inverses prop.
Of multiplication
For every nonzero real number a, there is
a unique 1/a
1
a 1
a
and
1
a 1
a
Properties
Identity property of addition
There is a unique real number 0 such that for
every real number a
a+0=a 0+a=0
Properties
Identity property of multiplication
There is a unique real number 1 such that for
every real number a, a 1 a and 1 a a
Properties
Multiplicative property of zero
a 0 0 and 0 a 0
Properties
Closure property of addition
For all real numbers a and b:
a+b is a unique real number
Properties
Closure property of Multiplication
For all real numbers a and b:
ab is a unique real number
Properties
Product of powers property
k k k
5
4
9
Properties
Power of a product property
(ab) a b
7
7 7
Properties
Power of a power property
(a ) a
2 4
8
Properties
Quotient of powers property
Subtract the exponents
5
x
3
x
2
x
Properties
Power of a quotient property
3
3
a
a
3
b
b
()
Properties
Zero Power
Property
(4ab) 1
0
Properties
Negative power property
a
2
1
2
a
Properties
Zero product property
If (x+3)(x-2)=0, then (x+3)=0 or (x-2)=0
Properties
Product of roots property
20 4 5
Properties
Quotient of roots property
45
3
5
Properties
Root of a power property
3
x
3
x
Properties
Power of a root property
( 7) 49 7
2
Solving first power inequalities in one
Variable
Examples of a first
power inequalities–
x5
When something is
equal to another number,
then you use a dark
circle, but when it isn’t
equal to, you use a a
non dark circle.
x2
5
2
Solving first power inequalities in one
Variable
Disjunction
–
A Disjunction uses the
word or
Example-
x 3 orx 1
1
3
Solving first power inequalities in one
Variable
Conjunctions
–
conjunctions include and
Example- x<3 and x>1
Or 3>x>1
1
3
Linear equations in two variables
Slope of lines
–
–
–
Horizontal: 0
Vertical: Undefined
Linear: rise over run
Linear equations in two variables
Equations of lines
–
–
–
–
Slope intercept form- Y=mx+b
Standard form: ax+by=c
vertical X= a constant
Horizontal y=a constant
Linear equations in two variables
In order to graph a line you need
–
–
–
A point and slope
Or two point
Or an equation
y 2x 1
slope
Y intercept
Y
intercept
Linear equations in two variables
How to find intercepts
–
–
X intercept- look for a point on the graph where y
equals zero
Y intercept- look for a point on the graph where x
equals zero
Linear equations in two variables
y y
How and when to use the point slope
formula–
You use the point slope formula when you don’t
know the y-intercept
Linear systems
Substitution Method–
Example-
Plug 15-x in for y
x 45 38
x 7
4x 45 3x 38
x y 15
4 x 3 y 38
x y 15
4 x 3(15 x) 38
y 15 x
Linear systems
Addition and
Subtraction
Method
(Elimination)
–
Example-
5 x y 12
3x y 4
Since the y’s
already
cross each
other out
there is no
need to use
the least
common
denominator
8 x 16 x 2
Linear systems
You can use
graphing but it only
gives an estimate
Linear systems
Check for understanding of terms–
–
–
Dependent system- Infinite set or all points (if
same line is used twice)
Inconsistent system-Null set (if they are parallel)
Consistent system-One point (if they cross)
Factoring
Methods
–
–
–
–
–
–
GCF- always look for the GCF first
Difference of Squares- used for binomials
Sum or Difference of cubes- used for binomials
PST- For trinomials
Reverse of FOIL- Trinomials
Grouping- Grouping
Factoring
GCF
–
Example
-
2x 8x 8
2
2( x 2 4 x 4)
2( x 2)( x 2)
Factoring
Difference of Squares
75 x 108 y
4
2
3(25 x 36 y )
4
2
3(5x – 6y) (5x 6y)
2
2
Factoring
Sum or difference of cubes
x3 y 3
( x y)( x xy y )
2
2
Factoring
Perfect Square Trinomial
x 4x 4
2
( x 2)
2
Factoring
Reverse Foil–
Trial and error
ax 2 bx c
( _ _ )( _ _ )
ax 2 bx c
(_ _)(_ _)
ax 2 bx c
(_ _)(_ _)
Factoring
Grouping–
Example-
b3 2b 2 ab 2a
b2 (b 2) a(b 2)
(b2 a)(b 2)
Rational expressions
Simplify by factor and cancel-
x x x( x 1)
x
x 1
x 1
2
Rational Expressions
Addition and Subtraction of rational
expressions
–
Addition-use LCM to cancel out the variable
a 2b 1
a 4b 5
6b 6
b 1
Rational Expressions
Subtraction of rational
expressions
–
–
Use LCM to cancel out the variablesExample-
6a 4b 5
6a 2b 1
2b 4
b2
6a 8 5
8 8
6 a 3
1
a
2
Rational Expressions
Multiplication and division of rational
expressions
–
Example-
2xy
2
3
4
4x y z
3
2 xy
2 xyz
Quadratic equations in one variable
Solve by factoring
–
Example
x2 2 x 8
( x 2)( x 4) 0
x 2 x 4
Quadratic equations in one variable
Solve by taking the square root of each side
–
Examplex 2 49 0
49
49
x 2 49
x7
Quadratic equations in one variable
Solve by completing the square
–
Examplex2 6x 2 0
Take half
of x and
square it
x 2 6 ____ 2 _____
x2 6x 9 2 9
( x 3) 2 11
x 3 11
Quadratic equations in one variable
Quadratic
formula
–
Example
Quadratic
Equation
b b 2 4ac
2a
x 2 3 x 10 0
3
3
9 (40)
2
49
2
3 7
5
2
or
37
2
2
Quadratic equations in one variable
b 2 4ac
What does the discriminant tell you?
–
Discriminant is the value of
b 4ac
2
Functions
What does f(x) mean?
–
–
–
–
F(x)= name of independent variable or argument
Usually equal to “Y”
Not all relations are functions (those that are
undefined)
Ex.
f ( x) 3x 1 y
2
Functions
range and domain of a function
Domain- set of all x values
Range- set of all y values
Ex. f (0) let x 0 Ex.(2) f ( x) 0 when y 0
f ( x) 5 x 10 x
f (0) (0, 0)
2
0 5 x 2 10 x
5 x( x 2) 0
x 0x 2
Functions
Ordered pairs
–
–
Ex. (1,1) (5,5)
Slope equals
5 1
1
5 1
y 1x b
1 1 b
b0
yx
Functions
Quadratic functions
How to graph a parabola
–
–
–
–
–
If A>0 then it opens up
If A<0 then it opens down
Vertex- is equal to a –b/2a to find x
Plug into f(x) to find y
Axis of symmetry- vertical through the vertex so
x= -b/2a
Functions
How to graph a parabola cont.
–
–
–
Y int. let x=0 or f (0)
X int. let y=0 or f (x) (0)
Factor and find solutions
Simplifying expressions with
exponents
A.) Product of powers
a a a
m
n
m n
3 4
ex.2 2 2
3
4
2
7
Simplifying expressions with
exponents
B.) quotient of powers
a a a
m
n
mn
Ex. 2 2 2
4
2
4 2
2 4
2
Simplifying expressions with
exponents
C.) Power of a Power
(a ) a
m n
mn
Ex. (2 ) 2 512
3 3
9
Simplifying expressions with
exponents
D.) Power of a Product
(ab) a b
m
m m
Ex. (2 x) 2 x 16 x
4
4
4
4
Simplifying expressions with
exponents
E.) Power of a Quotient
m
a m a
( ) m
b
b
2
4 2 4
16 1
Ex. ( ) 2
8
8
64 4
Simplifying expressions with radicals
A.) Root of a Power
x
x
B.) Power of a Root
3
3
x x
2
Ex. 7 7
2
Simplifying expressions with
radicals
C.) Rationalizing the Denominator
–
Use the multiplication identity property
7
2
7 2
Ex.
( )
2
2 2
Word Problems
Example 1–
A baseball game has 1200 people attending. Adult tickets are 5
dollars an student tickets are two dollars. The total amount of
money made a tickets was 3660 dollars. The visiting team is
entitled to half of the adult tickets sales. How much money does
the visiting team get?
x y 1200
y x 1200
5 x 2 y 3660
5 x 2 x 2400 3660
3 x 1260
x 420 adults
other school gets $1050
Word Problems
Example 2–
Al left MUHS at 10:30 AM walking 4 mi/hr. Bob left MUHS
at noon running to catch up with Al. If Bob overtakes Al at
1:30 PM how fast was he running.
Step 1- label variables
rate time distance
mi
4
Al
3 hrs 12 mi
hr
Bob
b
mi
hr
3
hrs
2
3
b mi
2
Step 3- solve for the variable
2
2 3
12 b
3
3 2
8b
Step 2- write an equation
Equal
distance
3
12 b
2
Step 4
Bob’s rate- 8
mi
hr
Word Problems
Example 3–
A serving of beef has 320 more calories than a serving of
chicken. The calories in 3 servings of beef is equal to the
calories in seven servings of chicken. Find the number of
calories in a serving of each meat.
chicken : c
beef : c 320
3c 960 7c
3c
960 4c
4
4
240 c
3c
3(c 320) 7c
chicken : 240 calories
beef : 560 calories
Word Problems
6 w 6 34 w
w
w
5w 6 34
6
6
Example 4–
5w 40is 3 cm less then twice the width.
The length of a rectangle
5
5
The perimeter is 34 cm
more
then the width. Find the
w8
length and width of the rectangle?
2w-3
w
6w 6 34 w
w
2w-3
6 w 6 34 w
w
w
5 w 6 34
6
6
5w
40
5
5
w8
8cm
13 cm
Line of Best fit or Regression line
You use to the line of best fit to estimate
what the average is for the data
Your TI-84 calculator can determine the line
of best fit for you
Line of Best fit or Regression line
What is the best fit line here?
Draw a line on the
graph if you want.