Created by Cal Larson

 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

 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

 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

Multiplication Property (of Equality) Example: If a = b, then a + c = b + c Example: If a = b, then ca = cb

Symmetric Property (of Equality) Transitive Property (of Equality) Example: If “a” is a real number, then a = a Example: If a = b, then b = a. Example: If a = b, and b = c, then a = c.

Associative Property of Multiplication Example: (a + b) + c = a + (b + c) Example: (ab)c = a(bc)

Commutative Property of Multiplication Example: a + b = b + a Example: ab = ba

Example: a(b + c) = ab + ac

Prop of Reciprocals or Inverse Prop. of Multiplication Example: -(a + b) = (-a) + (-b) Example: a 1/a = 1 and 1/a a = 1 Example: a 1/a = 1 and 1/a a = 1

Identity Property of Multiplication Example: If a + 0 = a, then 0 + a = a. Example: If a 1 = a, then 1 a = a.

Closure Property of Addition Closure Property of Multiplication Example: If a 0 = 0, then 0 a = 0. Example: a + b is a unique real number Example: ab is a unique real number

Power of a Product Property Power of a Power Property Example: a m a n = a m+n Example: (ab) m = a m b m Example: (a m ) n = a mn

Power of a Quotient Property Example: ( ) m =

Negative Power Property Example: If any number to the 0 power is 1 x 0 =1 Example: If an exponent is to a negative number then the number is the denominator over 1 X -5 = 1/x 5

Example: If ab = 0, then a = 0 or b = 0.

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:

Example: r 2 =s 2 r=s r=-s

 This means means x is greater than or equal to 5  This means x is less then or equal to 11  This means x is greater than to 15  This means x is less than -5  They are mostly the same however they will not be equal

 IF YOU MULTIPLY OR DIVIDE BY A NEGATIVE NUMBER THEN SWITCH THE SIGN!!!!!!!!  I.E. Divide by –x and switch the inequality sign

 To graph you have to make a line graph and make is so x is equal or greater than five.  There should be a dark dot for greater than or equal however my math program won’t let me do it  To graph you make the line graph so x will be smaller than -5

 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.

 This one might be a little weird  And  Null set

 Have fun with this one The answer is all real numbers

 This is not a fun unit I hated it and I’m sure you will hate it also, have fun 

 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

 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).  For example if the coordinates are 3,4 and 6,8  4-8/3-6  -4/-3  4/3

 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

 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

 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

 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2  The slope is 2

 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

 5x+y(0)=10  X=2  y intercept is 0,2

 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

 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

 X+2y=10  X+y=7  Multiply by -1  -x-y=-7  Then “add” the two equations  Y=3  X+6=10  X=4

 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

 2x+3y=15  3x+3y=12  -2x-3y=-15  3x+3y=12  X=-3  -6+3y=15  Y=7

 I will cover this briefly because it was our last unit  The sum/difference of cubes is (a+b) 3  (a+b)(a 2 +ab+b 2 )  The grouping 3 by 1 is (a+b) 2 +c 2  ((a+b)+c)((a+b)+c)  A perfect square trinomial is (x+b) 2  X 2 +b2+b 2

 Dots or difference of two squares  (x-5)(x+2)  x  The GCF is greatest common factor  15x 2 +15x+30  15(x 2 +x+2)  Grouping 2 by 2 is  x 2 +2x+x 3 +2x 2  X(x+2)+x 2 (x+2)  (x+x 2 )(x+2)

 A rational number is a number expressed as quotient of two integers  The denominator has to have a variable in it

 It is a lot easier than it seems  For addition just add the numerator and denominator and just simplify  For X 2 /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

 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)

 For strait factoring you set the equation to 0  X 2 +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=x 2  Take the square root of both sides and you get your answer  5=x

 Completing the square  X 2 -6x-3=0  X 2 -6x =3  Add (b/2) 2 to both sides  x 2 -6x+9=12  (x-3) 2 =12  Get the square root and simplify  X-3=2 Square root of 3

 It should b 2 but my math program won’t let me do that  X 2 +7x+10  -21/2  The discriminant tells me if the equation will work or not  The discriminant is b 2 -4ac

 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

 F(x)=x 2 +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)=x 2 +x-6  (x+3)(x-2)  X intercept is -3,2

 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

 Graph the points .3,40.6, ,60 2, ,80 5,90  The answer is Y=10.1x+44.1

 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

 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

 A car 20% off costs $60,000  How much does it cost normally? 75,000

 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