Transcript a + b

Created by
Cal Larson
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It is simple find out what X is equivalent to.
You can add, subtract, multiply and/or divide
REMEMBER WHAT YOU DO ON ONE SIDE
OF THE EQUATION YOU DO TO THE
OTHER!!!!!!!!!!!!!!!!!!!!!!!
5=x+3
x=2
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5x+3=4x
X=-3
1/2x+15=20
x+30=40
x=10
5(x+2)=15
5x+10=15
5x=5
X=1
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5x+6x+14=4x+7(x+2)
11x+14=4x+7x+14
11x+14=11x+14
The answer is x=all real numbers or everything
2/x=5
2=10x
1/5=x
Example:
If a = b, then a + c = b + c
Multiplication Property (of
Equality)
Example:
If a = b, then ca = cb
Example:
If “a” is a real number, then a = a
Symmetric Property (of Equality)
Example:
If a = b, then b = a.
Transitive Property (of Equality)
Example:
If a = b, and b = c, then a = c.
Example:
(a + b) + c = a + (b + c)
Associative Property of
Multiplication
Example:
(ab)c = a(bc)
Example:
a+b=b+a
Commutative Property of
Multiplication
Example:
ab = ba
Example:
a(b + c) = ab + ac
Example:
-(a + b) = (-a) + (-b)
Prop of Reciprocals or Inverse
Prop. of Multiplication
a•
Example:
1/a = 1 and 1/a • a = 1
Example:
If a + 0 = a, then 0 + a = a.
Identity Property of
Multiplication
If a •
Example:
1 = a, then 1• a = a.
If a •
Example:
0 = 0, then 0 • a = 0.
Closure Property of Addition
Example:
a + b is a unique real number
Closure Property of
Multiplication
Example:
ab is a unique real number
Example:
am • an = am+n
Power of a Product Property
Example:
(ab)m = ambm
Power of a Power Property
Example:
(am)n = amn
Power of a Quotient Property
Example:
a m
am
( b ) = bm
Example:
If any number to the 0 power is 1
x0=1
Negative Power Property
Example:
If an exponent is to a negative
number then the number is the
denominator over 1
X-5= 1/x5
Example:
If ab = 0, then a = 0 or b = 0.
Example:
a b  ab
Quotient of Roots Property
The square root of a divided by
the square root of b equals the
square root of a over b
Example:
r2=s2 r=s r=-s
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This means means x is greater than or equal to
5 x5
This means x is less then or equal to 11
x  11
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This means x is greater than to 15 15 x
This means x is less than -5 x  5
They are mostly the same however they will
not be equal
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IF YOU MULTIPLY OR DIVIDE BY A
NEGATIVE NUMBER THEN SWITCH THE
SIGN!!!!!!!!
I.E.  x  15
Divide by –x and switch the
inequality sign
x  15
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To graph x  5 you have to make a line graph
and make is so x is equal or greater than five.
5
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There should be a dark dot for greater than or
equal however my math program won’t let me
do it
To graph 5  xyou make the line graph so x
will be smaller than -5
5
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It is the same with just greater than or less than
but there is no black dot just a circle on the
graph
If there are two equations and you use the
word and then you shade in the overlapping
area or the line
If there are 2 equations and they have the word
or then you just graph the two on the same
line.
15  3x
15  3x
5  x
5
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This one might be a little weird
5 x
 5 x
 And
x 5
x 5
 Null set
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Have fun with this one
x  5 or x  3
The answer is all real numbers
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This is not a fun unit I hated it and I’m sure
you will hate it also, have fun 
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Y=mx+b is very simple
Y is the outcome m is the slope x is the input
and b is the y-intercept
Y=3x-5 is an example of Y=mx+b
Y is the output 3 is the slope and -5 is the y
intercept
A 3 slope means the point slides over 1 and up
3
The y intercept is where the line touches the y
axis
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The Y intercept will always start be 0,b
Y=mx+b is standard form
To find the slope for a straight line you need to
take the difference of the rise (X) over the
difference over the run (Y). x1  x 2 / y1  y 2
For example if the coordinates are 3,4 and 6,8
4-8/3-6
-4/-3
4/3
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The slope is 4/3
Point slope form is when you have the slope and
you have a point on the graph
Y-y1=m(x-x1)
If the slope is 2 and the point on the graph is 0,3
Y-0=2(x-3)
Y=2x-6
Now it is in standard form
Some problems will ask for it in standard form
while others will ask for it in point slope form
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How do you find the y and x intercepts?
6x+2y=12
To find the y intercept you set Y to 0 and solve
to find the x intercept set x to 0 and solve
6x+0y=12
X=2 the y intercept is 0,2
0x+2y=12
y=6
The x intercept is 6,0
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They give you the slope and y intercept!!
This allows you to find the equation of a line in
standard form
Example from the last problem
6/2=3 the slope is 3
Y=3x+2
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What is the slope and y intercept of the
equation Y=5x-3?
Slope is 5 and y intercept is 0,-3
Put this equation in standard form
The coordinates are -3,1 and -2,3
1-3/-3+2
-2/-1
2
The slope is 2
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Y+3=2(x-2)
Y+3=2x-4
Y=2x-1
Find the x and y intercepts for the equation
5x+2y=20
5(0)+2y=20
Y=10
x intercept is 10
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5x+y(0)=10
X=2
y intercept is 0,2
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In this unit of slideshows I will show you how
to solve equations with y and x as variables
The first method is the substitution method
This method works when in one part of the
equation has the coefficient of x or y = 1
2y+x=15
2y+3x=20
X=-2y+15
2y+3(-2y+15)=20
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2y-6y-45=20
-4y=-25
Y=25/4
Now enter y into the original equation
50/4+x=15
X=1 1/2
Next is the elimination method
You try to eliminate one variable by
multiplying so one variable is the opposite of
the other variable
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X+2y=10
X+y=7
Multiply by -1
-x-y=-7
Then “add” the two equations
Y=3
X+6=10
X=4
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X=y+2
2x+2y=10
2(y+2)+2y=10
4y+4=10
4y=10
Y=2.5
2.5+2=x
x=5.5
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2x+3y=15
3x+3y=12
-2x-3y=-15
3x+3y=12
X=-3
-6+3y=15
Y=7
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I will cover this briefly because it was our last
unit
The sum/difference of cubes is (a+b)3
(a+b)(a2+ab+b2)
The grouping 3 by 1 is (a+b)2+c2
((a+b)+c)((a+b)+c)
A perfect square trinomial is (x+b)2
X2+b2+b2
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Dots or difference of two squares
(x-5)(x+2)
x2-3-10
The GCF is greatest common factor
15x2+15x+30
15(x2+x+2)
Grouping 2 by 2 is
x2+2x+x3+2x2
X(x+2)+x2(x+2)
(x+x2)(x+2)
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A rational number is a number expressed as
quotient of two integers
The denominator has to have a variable in it
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It is a lot easier than it seems
For addition just add the numerator and
denominator and just simplify
For X2/x you simplify so the answer is just x
For addition or subtraction of two rational
expressions you make the signs one and just
continue
x/y+x/y=x+x/y+y
The same applies for subtraction
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It is the same thing as addition
(x/y)*(x/y)=(2x/2y)
Division is different first you do the reciprocal
of one number then you multiply them
(x/y)*(x/y)=(x/y)/(y/x)
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For strait factoring you set the equation to 0
X2+10x+25=0
(x+5)(x+5) the
You want to set the answer to zero so you make x
be the opposite of the constant
The answer is x=-5
Another way is taking the root of both sides
25=x2
Take the square root of both sides and you get
your answer
5=x
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Completing the square
X2-6x-3=0
X2-6x
=3
Add (b/2)2 to both sides
x2-6x+9=12
(x-3)2=12
Get the square root and simplify
X-3=2 Square root of 3
 b  bb  4ac / 2a
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It should b2 but my math program won’t let me
do that
X2+7x+10
-21/2
The discriminant tells me if the equation will
work or not
The discriminant is b2-4ac
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F(x) is the same thing as y
Remember not all relations are functions
The domain is the x and the range is the y in
functions
If you are given two points on a graph you just
do point slope formula
You graph a parabola just like any graph but
you have more variables and it looks like either
a hill or a valley
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F(x)=x2+2x+1
What are the x intercepts?
(x+1)(x+1)
The x intercepts are -1 and -1
Graph the following equation on loose leaf
then check on your calculator also find the y
intercepts
F(x)=x2+x-6
(x+3)(x-2)
X intercept is -3,2
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Linear Regression is when you have points on
a graph but you don’t have an equation
Your TI-84 calculator should help you with this
There should be a sheet of paper that will tell
you how to do it
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Graph the points
.3,40 .6,50 1.25,60 2,70 3.25,80 5,90
The answer is Y=10.1x+44.1
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Dan is 5 years older than Karl and Jim is 3
years older than Dan their total age is 58, how
old is Karl?
Karl is 15 years old
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Two people are on a see saw one weighs 150
pounds and is 2 feet away from the fulcrum the
other person weighs 100 pounds how far away
does he have to be from the fulcrum to balance
the seesaw
3 feet
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A car 20% off costs $60,000
How much does it cost normally?
75,000
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Joe owes $50,000 to the mob, they charge 30%
interest after a year if he pays it back in 3 years
how much will he owe?
Remember I=PRT
$95,000
Note to self never loan money from the mob