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 Multiplication Property (of Equality)Equality)

Example: Example: If a = b, then a + c = b + cIf a = b, then a + c = b + c

Example: Example: If a = b, then ca = cbIf a = b, then ca = cb

Symmetric Property (of Symmetric Property (of Equality)Equality)

Transitive Property (of Equality)Transitive Property (of Equality)

Example: Example: If “a” is a real number, then If “a” is a real number, then

a = aa = a

Example: Example: If a = b, then b = a.If a = b, then b = a.

Example: Example: If a = b, and b = c, then a = c.If a = b, and b = c, then a = c.

Associative Property of Associative Property of MultiplicationMultiplication

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

Example: Example: (ab)c = a(bc)(ab)c = a(bc)

Commutative Property of Commutative Property of MultiplicationMultiplication

Example: Example: a + b = b + aa + b = b + a

Example: Example: ab = baab = ba

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

Prop of Reciprocals or Prop of Reciprocals or Inverse Prop. of Inverse Prop. of Multiplication Multiplication

Example: Example: -(a + b) = (-a) + (-b)-(a + b) = (-a) + (-b)

Example: Example: a a • • 1/a = 1 and 1/a 1/a = 1 and 1/a • • a = 1a = 1

Identity Property of Identity Property of Multiplication Multiplication

Example: Example: If a + 0 = a, then 0 + a = a.If a + 0 = a, then 0 + a = a.

Example: Example: If a If a • • 1 = a, then 11 = a, then 1• a = a• a = a..

Closure Property of Closure Property of Addition Addition

Closure Property of Closure Property of Multiplication Multiplication

Example: Example: If If a a • 0 = 0• 0 = 0, then , then 0 • a = 0 • a =

00..

Example: Example: a + ba + b is a unique real number is a unique real number

Example: Example: ab ab isis a unique real numbera unique real number

Power of a Product Power of a Product Property Property

Power of a Power Power of a Power Property Property

Example:Example: aam m • • aann = a = am+nm+n

Example: Example: (ab)(ab)mm = a = ammbbmm

Example: Example: (a(amm))nn = a = amnmn

Power of a Quotient Property Power of a Quotient Property

Example: Example: ( )( )mm = =a

bam

bm

Negative Power Property Negative Power Property

Example:Example:If any number to the 0 power is If any number to the 0 power is

11xx00=1 =1

Example: Example: If an exponent is to a negative If an exponent is to a negative number then the number is the number then the number is the

denominator over 1denominator over 1XX-5-5= 1/x= 1/x55

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

Quotient of Roots PropertyQuotient of Roots PropertyThe square root of a divided The square root of a divided

by the square root of b by the square root of b equals the square root of a equals the square root of a

over b over b

Example: Example: a b ab

Example: Example: rr22=s=s2 2 r=s r=-sr=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

5x

11x

x155x

IF YOU MULTIPLY OR DIVIDE BY A NEGATIVE NUMBER THEN SWITCH THE SIGN!!!!!!!!

I.E. Divide by –x and switch the inequality sign

15 x

15x

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

5x

5

5

x 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.

x315 x315

x 5

5

x5 5xx 5 5x

Have fun with this one

The answer is all real numbers5x or 3x

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

21/21 yyxx

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 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)(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

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)

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 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

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 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

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

aacbbb 2/4

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

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)=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

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 .6,50 1.25,60 2,70 3.25,80 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