intro to algebra section 1 a more

8/14/2019 Intro to Algebra Section 1 a More

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INTRO TO ALGEBRAINTRO TO ALGEBRA

SECTIONSECTIONBy Alice ChichisanBy Alice Chichisan

MCCMCC

Fall 2007Fall 2007


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TOPICS:

Absolute Value

Rule of Signs

Exponents Rule

Square Roots

Scientific Notation

Evaluating and Simplifying Variable Expressions

One and Two Step Equations

Graphing in the Cartesian System

Writing Equations from words

Distance-Rate-Time Problems

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ABSOLUTE VALUE

DEFINITION:THE ABSOLUTE VALUE OF A NUMBER, , DENOTED

BY N REPRESENTS THE DISTANCE AWAY FROM ZERO

ON THE NUMBER LINE.

FOR EXAMPLE, 3 3 AND -3 3 SINCE BOTH

3 AND -3 ARE 3 UNITS AWAY FROM ZERO ON THE

REAL NUMBER LINE.

N

= =

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EXAMPLES

EVALUATE:

A) 2 1

B) 6+2 4 5C) - -2

+ =

==

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MULTIPLICATION AND

DIVISION OF REALNUMBERS

FOR MULTIPLICATION:

( )( ) ( )( )( ) ( )

( )( ) ( )( )( ) ( )

+ + = ++ =

+ = = +

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EXAMPLES

2*3 6

2*3 6

2*( 3) 6( 2)( 3) 6

=

=

=

=

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MULTIPLICATION AND

DIVISION OF REALNUMBERS For Division:

+ = +++ =

= +

= +

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EXAMPLES

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10 25

10 25

10 2

510 25

=

= =

=


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THE EXPONENT RULES

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5

5

Start with the definition:

* * *... , n times, where is called the BASE

and is called the EXPONENT.

For example: 2 2* 2*2* 2* 2 32

: 2 10

na a a a a a

n

CAUTION

=

= =


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EXPONENT RULES

The following rules are VERYIMPORTANT!

MEMORIZE THEM so that you canshare them with your children andgrand children

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*

0

( )

1, 0

1, 0

m n m n

mm n

n

m n m n

m

m

a a a

aa

a

a a

a aa

a a

+

=

=

=

=

=


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EXAMPLES:

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( ) ( )

3 4 7

4

4 5 15

2 23 2 2

4 3

0

2 *2 2 128

3 13 33 3

5 4 125 16 109 11881

2 3 16 27 1111

5 1 1

= =

= = =

= = =

= = =


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EXAMPLES (CONT)

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( )

( )( )

( )

( ) ( )

43 3*4 12

22 3 2 3 4 3 7

22

2 2 2 2

4 42 4 4 2 4 8

2 2 2 4096

4 *2 2 *2 2 *2 2

11 1 1

2 2 42

3 3 81

X X X X

XY X Y X Y

= = =

= = =

= = =

= =


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SQUARE ROOTS

Definition:

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2

2

2 2

reads: the square root of a,

where is a real number greater or equalto zero.

For example: 81 9 9 and -9

since 9 81 and (-9) 81 as well.

a

aa a=

= = +

= =


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WRITING SQUAREROOTS IN EXPONENT

NOTATION Lets start with an example:

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12

13 3

224 4

So, in general,a

b a b

x x

x x

x x

x x

=

=

=

=


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SCIENTIFIC

NOTATION It is a way of representing very largeor very small numbers using powers

of 10.

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0

1

2

3

4

Recall positive powers of 10:

10 1

10 10

10 10010 1,000

10 10,000

=

=

==

=


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SCIENTIFIC

NOTATION(2)

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-1

2

3

4

5

Recall the negative powers of 10:

10 0.110 0.01

10 0.001

10 0.0001

10 0.00001

=

=

=

=

=


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SCIENTIFIC

NOTATION(3) DEFINITION:

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4

a number written in scientific notation is expressed in the form:

*10 , where is an integer between 1 and 10.

For example, the number 32,456 written in scientific notation

looks like: 3.2456*10 .

na n


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SCIENTIFIC

NOTATION(4) Examples: Put the following numbers in

scientific notation:

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4

7

3

1

5

56,200 5.6*10

45,000,000 4.5*10

21 2.1*10

0.00462 4.62*10

0.212 2.12*10

0.0000789 7.89*10

==

=

=

=

=


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SCIENTIFIC

NOTATION(5) Examples:

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( ) ( ) ( ) ( )

( ) ( )

( ) ( )

5

4

4 3 4 3 1 0

8 3 11

4 8 4 3

1.778*10 0.00001778

1.3495*10 13,495

8*10 4*10 8*4 10 *10 32*10 3.2*10 3.2

2*10 3*10 6*10

6*10 8*10 48*10 4.8*10

=

=

= = = =

=

= =


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SCIENTIFIC

NOTATION(6) Examples:

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5( 5 ( 1)) ( 5 1) 4

1

1 ( 1 3) 4

3

35

2

94

5

9*103*10 3*10 3*10 0.0003

3*10

9.5*10 1.9*10 1.9*10 0.000195*10

2.6*101.3*10 0.000013

2*10

3*100.5*10 5000

6*10

+

= = = =

= = =

= =

= =

E L NG ND


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EVALUATING ANDSIMPLIFYING VARIABLE

EXPRESSIONS WE WILL BREAK THIS TOPIC INTWO:

FIRST: EVALUATE VARIABLEEXPRESSIONS

SECOND: SIMPLIFY VARIABLEEXPRESSIONS

AND THIRD (I KNOW ) WE SHALLCOMBINE THE TWO

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EVALUATING VARIABLEEXPRESSIONS

SOME TERMINOLOGY:

AN ALGEBRAIC EXPRESSION IS SIMPLY AMATHEMATICAL STATEMENT CONTAININGNUMBERS AND LETTERS. FOR EXAMPLE:

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2 33 ,2 4 ,5 6 7 ARE ALL ALGEBRAIC EXPRESSIONSX A B X Y + +


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EVALUATING VARIABLEEXPRESSIONS(2)

A TERM IS SIMPLY PART OF AN ALGEBRAICEXPRESSION, SUCH AS 4X,5 ETC.

A VARIABLE IS A LETTER WE USE TO REPRESENTUNKNOWN QUANTITIES; MOST COMMONLY USEDARE X,Y, Z OR A,B,C

A COEFFICIENTIS THE NUMBER LOCATED IN FRONTOF THE VARIABLE SUCH AS 4XY COEFFICIENT IS 4

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When you are asked to evaluate anexpression what you need to do is simplyto plug in the value of the given variable(s)and perform the indicated operations toobtain a numerical result.

Consider the following example (in the

next slide):

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2

2 2

Evaluate:

1. 3 1 3

3 (1) 3(3) 1 9 10

2. 8 5 2

8 (5)(2) 8 10 8 18

3. 9 3

9 9(3) (3) 27 9 36

a bif a andb

a b

mn form andn

mn

m m form

m m

+==

+=+==

==

===

=

===


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2 3

2

4 3 14. 2 1

4 3 1 4*2 3*1 1 8 3 1 11 1 10 52 2 2 2

3 45. 2, 4 1

2

x yif x and y

x

x yx

x y z if x y and z

x y z

+ = =

+ + + = = = = =

+

= = = +


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SIMPLIFYING VARIABLEEXPRESSIONS

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2 2

FIRST WE NEED SOME TERM INOLOGY:

LIKE TERM S ARE TERMS THAT C ONTAIN THE SAME VARIABLES

AND EACH C ORRESPON DING VARIABLE IS RAISED AT THE SAME POWER.FOR EXAMPLE: 4 3 ARE LIKE TERMS SINCE BOTH TERMS X Y AN D X Y CONTAIN

THE SAM E VARIABLES AND EACH CORRESPON DING V ARIABLE IS RAISED

TO T HE SAM E POW ER; THAT IS, X IS RAISEDTO THE SECOND POWER AND

Y IS RAISED TO T HE FIRST POW ER.


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SIMPLIFYING VARIABLEEXPRESSIONS(2)

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3 5 3 3 5

ANOTHER EXAM PLE:

-12XY 12 ARE NOT LIKE TERM S.

WHY NOT?

NOT ICE T HAT IN T HE FIRST TER M X IS RAISED TO THE FIRST POW E

W HILE IN TH E SECOND TERM X IS RAISED TOTHE THIRD POW ER.

Z A N D X Y Z


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IMPORTANT

WE CAN ONLY COMBINE LIKE

TERMS! THAT IS WE CAN ADD OR

SUBTRACT ONLY LIKE TERMS!

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For example:

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2 2 2

We can combine the following terms:

4x+3x=7x (just add the coefficients, 4 and 3 respectively)

Which terms can we combine in the following example?

2 3 5 3 6 1 3 5 1

Try the following exam

x y x y x y x x y x y+ + = +

3 3

ple:

12 14 2 5 3 14 5 xyz x y y x xyz x + + + +


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. 5( 1) 5 1

EX: 5(2a-6)-3(4a-9) 10 30 12 27 2 3

EX: 5(8 2) (5 3) 3 17 40 10 5 3 3 17

45 7 3 17

48 10

EX x x

a a a

j j j j j j

j j

j

+ =

= + =

+ + = + + =

= + =

= +


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Lets try one more example:

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: Simplify

3 7 8 12 11 15 3 7

EX

x xy y x y x y xy + + = +

SOLVING ONE AND TWO


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SOLVING ONE AND TWOSTEP LINEAR

EQUATIONS OBSERVATIONS: My explanations will be somewhat different than

the explanations you will find in your textbook if

you have one or than those you will find inMySkillsTutor. But I believe mine are better

As always use the explanation that best suitsyou

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SOLVING ONE AND TWO


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EQUATIONS(2) What is an equation? Simply put is just a mathematical or

algebraic expression, in which the equal signis involved.

The equal sign states that the left side ofthe equation is equal to the right side of the

equation. A=B, or LEFT=RIGHT

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SOLVING ONE AND TWO


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EQUATIONS(3) The idea behind solving any kind of equation is to solvefor the variable involved in the equation. We do that by

1. combining like terms together-numbers with numbersand variables with variables.

2. put the variable on one side of the equation and thenumbers on the other side.

3. divide both sides of the equation by the coefficientof the variable (if applicable).

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SOLVING ONE AND TWO


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EQUATIONS(4)

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Solve the equation for the given variable:

4 8

The question is what value should x takeso that both sides of the equation areequal.

There is an easy way to figure it out. Simply move the negative 4 o

x =

n the RIGHT side

of the equation. Why? Because then we we'll have x by itself (notice that the coefficient

for x is one).

ATTENTION!

Everytime we move a number or a variablefrom one side of the equation to the other,

the sign of that quantity changes from +to - or from - to +. It becomes the opposite of what it used to be.

So, for us , 4 8, becomes 4 8 12 x x or x = = + =

SOLVING ONE AND TWO


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EQUATIONS(5) ATTENTION: ALWAYS CHECKYOUR ANSWER!

SO, LETS DO THAT!

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The equation was 4 8 and we determined that the solution

is x=12.

Is it true?

We check the answer by plugging in the value of 12

into the original equation.

So, 4 8 becomes 12 4 8 which is a true stat

x

x

x

=

=

= = ement,hence we have the correct solution.

SOLVING ONE AND TWO


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EQUATIONS(6) How about another example? Well, here it is:

Solve for x: x-2=7

What do you do?

Move the negative 2 on the right side of theequation

What happens to (-2) once it crosses over tothe other side? (smile!)

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The sign changes from (-) to (+) and the equationturns into:

X=7+2 or x=9

Now, do you remember the original equation? X-2=7

We found the solution to be x=9

Is it true?

Lets check:

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So, we plug in THE ORIGINAL EQUATIONthe value of the solution, x=9, and wedetermine if the two sides of the equation

are equal: So, x-2 becomes 9-2=7

Both sides are equal, we are good and life is

good!

Lets try another one:

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Ex.

X-22=-14

X=-14+22

X=8 Check: 8-22=-14

Good job!

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Ex:

75=83+x

Oopsnow x is on the Right side of theequation!....Mamma mia!

Not to worrydo the same thing as before:just keep x where it is and just move the 83over to the left

So, 75-83=x Or x=-8

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Consider the followingexample:

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x)!1left withareweandoutcancel2s(the15

2

30

2

2x

:xoftcoefficienby thesidesbothdividingbySimply,

that?get tocan weHow

2x.notx1forvaluetheiswantwhat weNow,302

=

=

=

x

x


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Did you forget something?I know you did not!

Check!

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Ex.

Solve: 3x-2x-4=9

Combining like terms (that is the terms containing x)we obtain:

X-4=9

Next, same old story

X=9+4

Or, x=13

What are you going to do next?

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You got it! Check it! In theORIGINAL EQUATION!

3X-2X-4=9 and our solution (or thevalue we found for x) is x=13

So, 3(13)-2(13)-4=39-26-4=13-4=9

YES!

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HOW ABOUT ANOTHER


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HOW ABOUT ANOTHEREXAMPLE?

CHECK IT OUT! 3K-12+K+2=5+3K+2 OK, you can do this! Look, all you have to do is

to combine like terms(variables with variables

and numbers with numbers). Lets do it! 3K-12+K+2=5+3K+2

4k-10=3k+7

Next put the terms containing the variable on

one side of the equal sign and the numbers onthe other side.

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4k-10=3k+7

4k-3k=7+10

K=17

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one?

I know you are!

Here we go again!

3(2+5x)-(1+14x)=6

What should we do first? Distribute, youare right! So,

6+15x-1-14x=6

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Anyway, we finished solving the equation and now itstime to check if our work is correct

Original equation:

3K-12+K+2=5+3K+2 and solution k=17

3(17)-12+17+2=51-12+19=58 (left side)

5+3(17)+2=5+51+2=58 (right side)

Since left side=right side we are done, we haveperformed a great work of art and we should feel

really good about ourselves!

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OBSERVATION:

A NEGATIVE SIGN IN FRONT OF THEPARENTHESIS CHANGES THE SIGN OF

EVERYTHING IN THAT PARENTHESIS. FOR EXAMPLE,

a-(3a+4b-2c-d) becomes a-3a-4b+2c+d

Ok, lets get back to our equation:

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3(2+5x)-(1+14x)=6

6+15x-1-14x=6

X+5=6

X=6-5 X=1

Uhhhare we right? I dont know, lets check

But we are right, I know lets do it anyway

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Remember, every time you check, check it in the original equation:

3(2+5x)-(1+14x)=6 and our solution is x=1

3(2+5*1)-(1+14*1)=3(2+5)-(1+14)=3*7-15=

=21-15=6 (this is the left side, remember?)

Again, since the left side of the equation equals the right side, we aregooood!

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Ready for another one?

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numbers.withnumbers

ables,with varivariablesbefore,asSame

this!dotohowknowWILLYOU

!courageousbebutfractions,know,I

5

817

5

3 xx=+


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175

8x3x

:oneintofractionstwothecombinecanwe

rdenominatosamethehaveweSince

175

85

3

5

817

5

3

=

=

=+

xx

xx


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obtainweand(-1)bysidesbothMultily

-17x-

:getweand5sout theCancel

1755

175

8x3x

=

=

=

=

x


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2.272.27

2.27:172.10:5

13617

5

51

5

17*817

5

17*3,

17xissolutionourand

5

817

5

3

check!oRemember t

=

=+

=+

=+

==+

rightsideleftside

So

xx


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Again, we did great! The left sideequals the right side, we are goood!

Life is good!

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Another example2 153 3

215 0

3 3215

3 33

15

315

x x

x x

x x

x

x

+ =

+ + =

+ =

=

=


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And another

4 622

7 7

4 6

227 7

222

7

x x

x x

x

+ =

=

=


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And another

(2):2 22

7 1( 2 )*1 (7)( 22)

2 154

2 154

2 277

a c Note ad bc

b d

x

x

x

x

x

= =

=

= =

= =


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Dont forget to check!

The original equation is

75=83+x and our solution is x=-8 75=83+(-8)=83-8

75=75

Boy, are we good!

Move this slide back

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R d f b k?


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Ready for a break?Me too

When we come back well look at more equationsthat require one more step

The fun never stops

Did you know that the Egyptians were able tosolve these types of equations without the aid ofany calculator? Now, they had a very weird andcumbersome way of solving, but they did it!

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Ok, here we go again

I dont know if youve noticed, but in all of theprevious equations the coefficient of the variable wasalways 1 (thats the number in front of the variable).

So weve always had the case where x=whatever.

But what happens if we have something like 3x=10?What do we do then?

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The idea is to solve for one x.

That is, we want the coefficient of x

to be equal to 1.

In our case, the coefficient of 3x is 3.So, what do we do?


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So, we have:

3 10

Divide both sides by 3.This results in:

3x 10

3 3

10This results in

3

Remember to check your answer. :)

x

x

=

=

=

Li ti ti


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Linear equations practice

problems


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GRAPHING IN THE

CARTESIAN SYSTEM The Cartesian System is just the x-ycoordinate system.

It is also called the Rectangular system(since it kind of looks like a rectangle).

I prefer to call it the Cartesian system inhonor of Renee Descartes who discovered

it.


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Renee Descartes Lived around the 1600. The legend of the discovery of the Cartesian

system

Negative numbers were not accepted at that time Do you know what else Descartes is known for?

Besides being a brilliant mathematician, he excelledat poetry and philosophy.

O d d P i d


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Ordered Pairs and

Graphs The graph of a point in two variablesxand y is the ordered pair

(x, y), where x and y are the coordinates of the point.

X, the first number, tells how far to the left or right the point isfrom the vertical axis.

Y, the second number, tells how far up or down the point is fromthe horizontal axis.

For instance, the point (1, 3) is the point that is 1 to left ofvertical axis and is three up from horizontal axis.


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y

x1

2

3 2 3X-axis

y-axis

origin

Quadrant IQuadrant II

Quadrant IIIQuadrant IV

(x, y) coordinate system terms

(3,2)(-3,2)

(-3,-3)

(1,-2)


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MORE ON:

The Rectangular CoordinateSystem


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Rectangular Coordinate System

or Cartesian Plane

otting points on aR l C di


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g pRectangular Coordinate

System

otting points on aR l C di


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System

otting points on aR t l C di t


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System

Example: Create a table of values for each


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Example: Create a table of values for eachequation, and graph it on grid paper

72 =+ yx

5= yx

x Original equation (x, y)

otting points on aR t l C di t


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System

G id li f if i


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Guidelines for verifyingsolutions

Substitute the values of x and y into theequation.

Simplify each side of the equation. If each side simplifies to the same number,the ordered pair is a solution. If the twosides yield different numbers, the orderedpair is not a solution.

V if i l i f


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Verifying solutions of anEquation

Determine whether each ordered pair is asolution of the given equation.

y = 5x - 7; (2, 3), (1, 5), (-1, -12)


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The Distance Formula

Given the two points (x1,y1) and(x2,y2), the distance between these

points is given by the formula:


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The Distance FormulaFind the distance between the points (2, 3) and (4, 4).

Just plug them in to the Distance Formula:


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Example: Show that the points(1,2),(3,1) and (4,3)

are vertices of a right angle triangle.

Determine whether the set ofi t i lli


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points is collinear.{A(2,6),B(5,2),C(8,-2)}


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Midpoint Formula Find the midpoint between (6.4, 3) and (10.7, 4).

Apply the Midpoint Formula:

So the answer isP= (2.15, 3.5)


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Midpoint Formula Technically, the Midpoint Formula is the

following:


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Section 3.2

Graphs of Functions


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STANDARD FORM


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y

x

(x,0)(x,0)

Standard form of a linear equation is Ax + By = C. A and B are

not both zero. A quick way to graph this form is to plot itsintercepts (when they exist).Draw a line through the two points.

(x,0)(x,

Ax +By = C

The x-intercept is thex-coordinate of the pointwhere the line intersects

thex-axis.

Ax +By = C

Drawing Quick Graphs


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Let y= 0.

2x + 3y = 12 Standard form.

2x + 3(0)= 12

x = 6

Draw a line through the two points.

Solve for x.

Graph 2x + 3y = 12

SOLUTIONMETHOD 1:USE STANDARD FORM

2(0) + 3y = 12 Let x= 0.

Solve for y.y = 4

The x-intercept is 6, so plot the point (6, 0).

(6, 0)

(0, 4)

The y-intercept is 4, so plot the point (0, 4).


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Graphs of non linear equations(Parabola Graph)

Sketch the graph of x2+y=0Steps: Isolate y

Create a table of values

Plot the points & connect them

x Y = -x2 (x, y)

-2

-1

0

1

2

3

STANDARD FORM


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GRAPHING LINEAR EQUATIONS IN STANDARD FORM

Write equation in standard form.

Find x-intercept by letting y= 0. Solve for x. Usex-intercept to plot point where line crosses x-axis.

Find y-intercept by letting x= 0. Solve for y. Usey-intercept to plot point where line crosses y-axis.

Draw line through points.

The standard form of an equation gives you a quickway to graph the equation.

1

2

3

4

Graphs of an Absolute Value


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equations(V Graph)

Sketch the graph of y = |x - 1|Steps: Isolate y

Create a table of values

Plot the points & connect them

x y = |x - 1| (x, y)

-2

-1

0

1

2

3


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Section 3.3

Slope and Graphs of

Linear Equations


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Definition of Slope

2 1

2 1

y y changeiny rise

x x changeinx run

= =

(4,1)

(0,

2)

3

4

Rise = 3

Run = 4

Slope =


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Slope of a Line A line with positive slope (m > 0) rises from

left to right.

A line with negative slope (m < 0) falls fromleft to right.

A line with zero slope (m = 0) is horizontal.

A line with undefined slope is vertical.

STANDARD FORM


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The equation of a vertical line cannot be written in slope-intercept

form because the slope of a vertical line is not defined. Everylinear equation, however, can be written in standard form

even the equation of a vertical line.

HORIZONTAL AND VERTICAL LINES

HORIZONTAL LINES The graph of y = c is a horizontal line

through (0, c).

VERTICAL LINES The graph of x= c is a vertical line

through (c, 0).

Graphing Horizontal and Vertical Lines


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Graphy = 3 andx = 2

SOLUTION

The graph of y= 3 is a horizontal line

that passes through the point (0, 3).

Notice that every point on the line has

ay-coordinate of 3.

(0, 3)

The graph ofx = 2 is a vertical line that

passes through the point ( 2, 0). Notice

that every point on the line has anx-coordinate of 2.

(2, 0)

y= 3

x =2

SLOPE-INTERCEPT FORM


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y

xm is the slope

b is they-intercept

The slope intercept form

of a linear equation is

y = mx + b.

(0 , b)

If the graph of an equation intersects they -axis at the point

(0, b), then the number b is they -intercept of the graph.

To

find they -intercept of a line, letx = 0 in an equation for the

line and solve fory.

y = mx + b

SLOPE-INTERCEPT FORM


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GRAPHING EQUATIONS IN SLOPE-INTERCEPT FORM

The slope-intercept form of an equation gives you a quickway to graph the equation.

Find y-intercept, use it to plot point where line crossesy-axis.

Find slope, use it to plot a second point on line.

Draw line through points.

Write equation in slope-intercept form by solving for y.STEP 1

STEP 2

STEP 3

STEP 4


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Sketching a graph If necessary rewrite the equation in slope-

intercept form and identify the slope Find the x and/or y intercepts of the

equation Graph the points representing the intercepts

and use the slope to include at least threeother points on the graph.


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Parallel and Perpendicular Lines Two distinct non-vertical lines are parallel if

and only if they have the same slope. Two distinct non-vertical lines are

perpendicular if and only if they have theirslopes are negative reciprocals of each otheri.e.m1 = -or m1 m2 = -12

1

m


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Slope as a Rate of Change Slope is often used to describe a constant

rate of change or an average rate of change


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Section 3.4

Equations of Lines

Point Slope Form of the


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Point Slope Form of theEquation of a Line

Write the equation of a line given the slope(m) and one point can be done using theslope formula in proportion with the slope

The point-slope formula of the equation ofa line is most useful:

y y1

= m ( x x1

)


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Application Linear Extrapolation: when the estimated

point lies to the right of the given points

Linear Interpolation: when the estimatedpoint lies between two given points

Equation of a Line Passing


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Equation of a Line PassingThrough Two (2) Points

1. Find the slope of the points using the slopeformula

2. Insert the slope and either of the points into the

point-slope form of the equation of a line andsimplify

3. Either leave the equation in standard (general)form ( ax+by+c=0) or slope intercept form (y =mx+b )


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Lines for graphing < or > : graph uses a dotted or broken

line

or : graph uses a solid line


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Section 3.5

Graphs of Linear

Inequalities


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Verifying solutions Solutions for linear inequalities are

verified the same as in linear

equations: by substituting the valuesinto the inequality and checking to seeif the resulting statement is true.


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Graphing Method Solve the inequality in two variables

for the dependent variable (usually they variable) if necessary.

It is done the same as changing anequation in standard form into slope-intercept form.


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Graphing Method Make a table of x and y values for the

graph using the inequality as anequation.

Graph at least 5 ordered pairsremembering to use the type of linedictated by the inequality symbol.


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Graphing Method If the inequality is greater than or greater

than or equal to (> or ) shade above theline.

If the inequality is less than or less than orequal to (< or ) shade below the line.


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Key words for inequalities More than: > Less than:

intro to algebra section 1 a more

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