Algebra 1

Algebra 1Algebra 1Marcos De la Marcos De la

CruzCruzAlgebra 1Algebra 1((66thth period period))

Ms.HardtkeMs.Hardtke

5/14/105/14/10

Algebra TopicsAlgebra Topics

1- 1- PropertiesProperties

2- Linear Equations2- Linear Equations3- Linear Systems3- Linear Systems4- Solving 14- Solving 1stst Power Equations (1 Variable) Power Equations (1 Variable)5- Factoring5- Factoring6- Rational Expressions6- Rational Expressions7- Quadratic Equations7- Quadratic Equations8- Functions8- Functions9- Solving 19- Solving 1stst Power Inequalities (1 Variable) Power Inequalities (1 Variable)10- Word Problems10- Word Problems11- Extras11- Extras

Addition Property of EqualityAddition Property of Equality

If the same number is If the same number is added to both sides of added to both sides of an equation, both sides an equation, both sides will be and remain equalwill be and remain equal 3=3 (equation)3=3 (equation)

If 2 is added to both sidesIf 2 is added to both sides 2+3=3+22+3=3+2 5=55=5

Negative Special CaseNegative Special Case y=3x+5 (equation)y=3x+5 (equation)

If (-3) is added to both If (-3) is added to both sidessides

Y-3=3x+5-3Y-3=3x+5-3 Y-3=3x+2Y-3=3x+2 Its still equalIts still equal

Multiplication Property of EqualityMultiplication Property of Equality

States that when both sides of an equal States that when both sides of an equal equation is multiplied and the equation equation is multiplied and the equation remains equalremains equal If 5=5 (equation)If 5=5 (equation)

5x2=2x55x2=2x5 You multiply both sides by 2You multiply both sides by 2

10=1010=10 Still remains equalStill remains equal

Reflexive Property of EqualityReflexive Property of Equality

When something is the When something is the exact same on both exact same on both sidessides A = AA = A 7x = 7x7x = 7x 3456x = 3456x3456x = 3456x

Symmetric Property of EqualitySymmetric Property of Equality

When two variables are different but are the When two variables are different but are the same number/amount same number/amount (equal symmetry)(equal symmetry)

If a=b, then b=aIf a=b, then b=a If c=d, then d=cIf c=d, then d=c If xyp=xyo, then xyo=xypIf xyp=xyo, then xyo=xyp

Transitive Property of EqualityTransitive Property of Equality

When numbers or variables are all equalWhen numbers or variables are all equal If a=b and b=c, then c=aIf a=b and b=c, then c=a if 5x=100 and 100=4y, then 4y=5xif 5x=100 and 100=4y, then 4y=5x if 0=2x and 2x=78p, then 78p=0if 0=2x and 2x=78p, then 78p=0

Associative Property of AdditionAssociative Property of Addition

The sum of a set of numbers is the same no The sum of a set of numbers is the same no matter how the numbers are grouped. matter how the numbers are grouped. Associative property of addition can be Associative property of addition can be summarized algebraically as:summarized algebraically as:

(a + b) + c = a + (b + c)(a + b) + c = a + (b + c) (3 + 5) + 2 = 8 + 2 = 10(3 + 5) + 2 = 8 + 2 = 10

3 + (5 + 2) = 3 + 7 = 103 + (5 + 2) = 3 + 7 = 10 (3 + 5) + 2 = 3 + (5 + 2). (3 + 5) + 2 = 3 + (5 + 2).

Associative Property of MultiplicationAssociative Property of Multiplication

The product of a set of numbers is the same no The product of a set of numbers is the same no matter how the numbers are grouped. The matter how the numbers are grouped. The associative property of multiplication can be associative property of multiplication can be summarized algebraically as: summarized algebraically as:

(ab)c = a(bc)(ab)c = a(bc)

Commutative Property of AdditionCommutative Property of Addition

The sum of a group of numbers is the same The sum of a group of numbers is the same regardless of the order in which the numbers regardless of the order in which the numbers are arranged. Algebraically, the commutative are arranged. Algebraically, the commutative property of addition states: property of addition states:

a + b = b + aa + b = b + a 5 + 2 = 2 + 5 because 5 + 2 = 7 and 2 + 5 = 75 + 2 = 2 + 5 because 5 + 2 = 7 and 2 + 5 = 7 -3 + 11 = 11 - 3-3 + 11 = 11 - 3

Commutative Property of Commutative Property of MultiplicationMultiplication

The product of a group of numbers is the same The product of a group of numbers is the same regardless of the order in which the numbers regardless of the order in which the numbers are arranged. Algebraically, commutative are arranged. Algebraically, commutative property of multiplication can be written as:property of multiplication can be written as:

ab = baab = ba 8x6 = 48 and 6x8 = 48: thus, 8x6 = 6x8 8x6 = 48 and 6x8 = 48: thus, 8x6 = 6x8

Distributive PropertyDistributive Property

The sum of two addends The sum of two addends multiplied by a number multiplied by a number is the sum of the is the sum of the product of each addend product of each addend and the number and the number A(b+c)A(b+c)

Ab + AcAb + Ac

3x(2y+4)3x(2y+4) 6xy + 12x6xy + 12x

Property of Opposites/Inverse Property Property of Opposites/Inverse Property of Additionof Addition

When a number is added by itself negative or When a number is added by itself negative or positive to make zeropositive to make zero

a + (-a) = 0a + (-a) = 0 5 + (-5) = 05 + (-5) = 0 -3y + (3y) = 0-3y + (3y) = 0

Property Of Reciprocals/Inverse Property Of Reciprocals/Inverse Property of MultiplicationProperty of Multiplication

For two ratios, if For two ratios, if a/b = a/b = c/dc/d, then , then b/a = d/cb/a = d/c a(1/a) = 1a(1/a) = 1 5(1/5) = 15(1/5) = 1 8/1 x 1/8 = 1 8/1 x 1/8 = 1

A number times its A number times its reciprocal, always reciprocal, always equals oneequals one

A A Reciprocal Reciprocal is its is its reverse and opposite reverse and opposite

(the signs switch from (the signs switch from ++ to to —— or vice versa) or vice versa)

Reciprocal FunctionReciprocal Function (continued)(continued)

The reciprocal function: The reciprocal function: yy = 1⁄ = 1⁄xx. For every . For every xx except 0, except 0, yy represents its multiplicative inverse represents its multiplicative inverse

Identity Property of AdditionIdentity Property of Addition

A number that can be added to any second A number that can be added to any second number without changing the second number. number without changing the second number. Identity for addition is 0 (zero) since adding Identity for addition is 0 (zero) since adding zero to any number will give the number itself:zero to any number will give the number itself:

0 + a = a + 0 = a0 + a = a + 0 = a 0 + (-3) = (-3) + 0 = -30 + (-3) = (-3) + 0 = -3 0 + 5 = 5 + 0 = 5 0 + 5 = 5 + 0 = 5

Identity Property of MultiplicationIdentity Property of Multiplication

A number that can be multiplied by any A number that can be multiplied by any second number without changing the second second number without changing the second number. Identity for multiplication is "1,“ number. Identity for multiplication is "1,“ instead of 0, because multiplying any number instead of 0, because multiplying any number by 1 will not change it.by 1 will not change it.

a x 1 = 1 x a = aa x 1 = 1 x a = a (-3) x 1 = 1 x (-3) = -3(-3) x 1 = 1 x (-3) = -3 1 x 5 = 5 x 1 = 5 1 x 5 = 5 x 1 = 5

Multiplicative Property of ZeroMultiplicative Property of Zero

Anything number or Anything number or variable multiplied variable multiplied times zero (0), will times zero (0), will always equal zeroalways equal zero 5 x 0=05 x 0=0 5g x 0=05g x 0=0

No matter what number is No matter what number is being multiplied by zero, being multiplied by zero, it will it will alwaysalways be zero be zeroA really long way to explain the A really long way to explain the Multiplicative Property of ZeroMultiplicative Property of Zero

(Proof)(Proof)

http://upload.wikimedia.org/math/b/5/8/b5892630f1d2f28a580331a1d7e3e79f.png

Closure Property of AdditionClosure Property of Addition

Sum (or difference) of 2 real numbers equals a real Sum (or difference) of 2 real numbers equals a real number number 10 – (5)= 510 – (5)= 5

Closure Property of MultiplicationClosure Property of Multiplication

Product (or quotient if denominator 0) of 2 Reals Product (or quotient if denominator 0) of 2 Reals equals a real numberequals a real number 5 x 2 = 105 x 2 = 10

(Exponents) Product of Powers (Exponents) Product of Powers PropertyProperty

ExponentsExponents Exponents are the little Exponents are the little

numbers above numbers above numbers, that mean that numbers, that mean that the number is multiplied the number is multiplied by itself that many timesby itself that many times 7 × 7 =7 × 7 =(7 × 7) × (7 × 7 × 7 × 7 × 7 × 7) (7 × 7) × (7 × 7 × 7 × 7 × 7 × 7)

When two exponents or When two exponents or numbers with exponents numbers with exponents are being multiplied, are being multiplied, you add both exponents, you add both exponents, but you still multiply but you still multiply the number or variablethe number or variable 3x (5x ) =3x (5x ) =

15x15x 15x15x

22 66 33 44

(3+4)(3+4)

77

Power of a Product PropertyPower of a Product Property

To find a power of a product, find the power To find a power of a product, find the power of each factor and then multiply. In general:of each factor and then multiply. In general:

((abab) = ) = aa · · bb OrOr

a a · · b b = ( = (abab)) (3(3tt))

(3(3tt) = 3 · ) = 3 · tt = 81 = 81tt

mmmmmm

mm mm mm

44

44 44 44 44

Power of a Power PropertyPower of a Power Property

To find a power of a power, multiply the To find a power of a power, multiply the exponents. (Its basically the same as the exponents. (Its basically the same as the Power of a Product PropertyPower of a Product Property, if forgotten, go , if forgotten, go one slide back and review.)one slide back and review.) (5 )(5 )

(5 )(5 )(5 )(5 ) = 5 5(5 )(5 )(5 )(5 ) = 5 5

Its basically this:Its basically this:

(a ) = a(a ) = a

33 44

33 33 33 33 3(4)3(4) 1212

bb cc bcbc

Quotient of Powers PropertyQuotient of Powers Property When both the denominator and When both the denominator and

numerator of a fraction have a numerator of a fraction have a common variable, it can be common variable, it can be canceled, therefore not usable canceled, therefore not usable anymoreanymore

Also when a variable is canceled, Also when a variable is canceled, the exponents are subtracted, the exponents are subtracted, instead of added as in the instead of added as in the Product of Powers PropertyProduct of Powers Property

a /a aa /a a

5 5x5x55 5x5x5 —— —————— —— ——————

5 5x55 5x5

= 5= 5

(the canceling of common (the canceling of common factors)factors)

bb cc b-cb-c

33

22

Power of a Quotient PropertyPower of a Quotient Property

This is almost the same as the This is almost the same as the Quotient of Quotient of Powers PropertyPowers Property, but this time, an entire , but this time, an entire fraction is multiplied by an exponentfraction is multiplied by an exponent

You also have to cancel the common factors, if You also have to cancel the common factors, if there are anythere are any (a/b) a /b — (a/b) a /b — (a/6)(a/6)

(and vice versa) (a /36)(and vice versa) (a /36)

cc cccc 22

22

Zero Power PropertyZero Power Property

If a variable has an exponent of zero, then it If a variable has an exponent of zero, then it must equal onemust equal one a =1a =1 b =1b =1 c b a =1c b a =1 (a ) =1(a ) =1

0

0

0 0 0

2 0

Negative Power PropertyNegative Power Property

When a fraction or a number has negative When a fraction or a number has negative exponents, you must change it to its reciprocal exponents, you must change it to its reciprocal in order to turn the negative exponent into a in order to turn the negative exponent into a positive exponentpositive exponent 4 ¼ 1/164 ¼ 1/16

-2-2 22

The exponent turned from negative to positiveThe exponent turned from negative to positive

Zero Product PropertyZero Product Property

When both variables equal zero, then one or When both variables equal zero, then one or the other must equal zerothe other must equal zero if ab=0, then either a=0 or b=0if ab=0, then either a=0 or b=0 if xy=0, then either x=0 or y=0if xy=0, then either x=0 or y=0 if abc=0, then either a=0, b=0, or c=0if abc=0, then either a=0, b=0, or c=0

Product of Roots PropertyProduct of Roots Property

The product is the same as the product of The product is the same as the product of square rootssquare roots

Quotient of Roots PropertyQuotient of Roots Property

The square root of the quotient is the same as The square root of the quotient is the same as the quotient of the square roots:the quotient of the square roots:

AA AA

BB BB

Root of a Power PropertyRoot of a Power Property

Power of a Root PropertyPower of a Root Property

Density Property of Rational NumbersDensity Property of Rational Numbers

Between any two rational numbers, there Between any two rational numbers, there exists at least one additional rational numberexists at least one additional rational number

1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9

4.54.5 oror

44½½

WebsitesWebsites

PROPERTIESPROPERTIESϮ http://www.my-ice.com/ClassroomResponse/1-6.htmhttp://www.my-ice.com/ClassroomResponse/1-6.htmϮ http://intermath.coe.uga.edu/dictnary/descript.asp?termID=300http://intermath.coe.uga.edu/dictnary/descript.asp?termID=300Ϯ http://en.wikipedia.org/wiki/Multiplicativeinversehttp://en.wikipedia.org/wiki/Multiplicativeinverse Ϯ http://ask.reference.com/related/Reciprocal+of+a+Number?qsrc=2892&l=dir&o=10601http://ask.reference.com/related/Reciprocal+of+a+Number?qsrc=2892&l=dir&o=10601 Ϯ http://faculty.muhs.edu/hardtke/Alg1_Assignments.htmhttp://faculty.muhs.edu/hardtke/Alg1_Assignments.htm — MUHS— MUHSϮ http://www.northstarmath.com/sitemap/MultiplicativeProperty.htmlhttp://www.northstarmath.com/sitemap/MultiplicativeProperty.html Ϯ http://hotmath.com/hotmath_help/topics/product-of-powers-property.htmlhttp://hotmath.com/hotmath_help/topics/product-of-powers-property.html Ϯ http://hotmath.com/hotmath_help/topics/power-of-a-product-property.htmlhttp://hotmath.com/hotmath_help/topics/power-of-a-product-property.html Ϯ http://hotmath.com/hotmath_help/topics/power-of-a-power-property.htmlhttp://hotmath.com/hotmath_help/topics/power-of-a-power-property.html Ϯ http://hotmath.com/hotmath_help/topics/quotient-of-powers-property.html http://hotmath.com/hotmath_help/topics/quotient-of-powers-property.html Ϯ http://hotmath.com/hotmath_help/topics/power-of-a-quotient-property.html http://hotmath.com/hotmath_help/topics/power-of-a-quotient-property.html Ϯ http://hotmath.com/hotmath_help/topics/properties-of-square-roots.htmlhttp://hotmath.com/hotmath_help/topics/properties-of-square-roots.html Ϯ http://www.slideshare.net/misterlamb/notes-61 http://www.slideshare.net/misterlamb/notes-61 Ϯ http://www.ecalc.com/math-help/worksheet/algebra-help/ http://www.ecalc.com/math-help/worksheet/algebra-help/

Hotmath.com


1- Properties1- Properties

2- Linear Equations2- Linear Equations3- Linear Systems3- Linear Systems4- Solving 14- Solving 1stst Power Equations (1 Variable) Power Equations (1 Variable)5- Factoring5- Factoring6- Rational Expressions6- Rational Expressions7- Quadratic Equations7- Quadratic Equations8- Functions8- Functions9- Solving 19- Solving 1stst Power Inequalities (1 Variable) Power Inequalities (1 Variable)10- Word Problems10- Word Problems11- Extras11- Extras

Standard/General FormStandard/General Form

Standard FormStandard Form Ax + By = CAx + By = C The terms A, B, and C are The terms A, B, and C are

integers (could be either integers (could be either positive or negative positive or negative numbers or fractions)numbers or fractions)

If If FractionsFractions:: Multiply each term in the Multiply each term in the

equation by its equation by its LCD (Lowest LCD (Lowest Common Denominator)Common Denominator)

Either add or subtract to get Either add or subtract to get either X or Y isolated, in one either X or Y isolated, in one side of the =side of the =

If If DecimalsDecimals:: Multiply each term in the Multiply each term in the

equation depending on the equation depending on the decimal with the most decimal with the most numbers (by 10, 100, 1000, numbers (by 10, 100, 1000, etc)etc)

1.23 (multiply times 100)1.23 (multiply times 100) 123.00123.00

Subtract or add to get X or Y Subtract or add to get X or Y isolatedisolated

If If Normal NumbersNormal Numbers (neither (neither fractions or decimals):fractions or decimals): Just add or subtract to get X Just add or subtract to get X

or Y isolatedor Y isolated

Graph PointsGraph Points

A Graph Point contains of an X and a YA Graph Point contains of an X and a Y

(x,y)(x,y) The X and Y mean where exactly the point is The X and Y mean where exactly the point is

locatedlocated

X line graphX line graph

Y line graphY line graph

Standard/General Form Ex.Standard/General Form Ex.

Fractions:Fractions: You multiply by the You multiply by the LCMLCM

Which in this case is Which in this case is 20x20x

Then to double check Then to double check it…it…

Point-Slope FormPoint-Slope Form The Point-Slope form. got its The Point-Slope form. got its

name because it uses a single name because it uses a single point in a graph and a on the point in a graph and a on the slope of the lineslope of the line

It is usually used to find the slope It is usually used to find the slope of a graph, if the slope is not of a graph, if the slope is not given in a certain problem or given in a certain problem or equationequation

The Y on the Point-Slope form., The Y on the Point-Slope form., doesn’t mean that the Y is doesn’t mean that the Y is multiplied by one, but it means to multiplied by one, but it means to use the first Y of the two or one use the first Y of the two or one point given as a problem (same point given as a problem (same with X)with X) (4,3) and the slope is 2(4,3) and the slope is 2

M = slopeM = slope Y—stays the sameY—stays the same X —is 4 (because 4 is in X —is 4 (because 4 is in

the x spot)the x spot) Y — is 3Y — is 3 X—stays the sameX—stays the same

If the problem gives you two points If the problem gives you two points and no slope, then you are free to and no slope, then you are free to choose what which or the Xs or the choose what which or the Xs or the Ys you may want to use for your Ys you may want to use for your Point-Slope Form.Point-Slope Form.

11

11

11

exex

Point-Slope FormPoint-Slope Form

(4,3) and m=2(4,3) and m=2 you must convert “it” to a you must convert “it” to a slope-intercept formslope-intercept form

Y=Mx + BY=Mx + B Y-3 = 2(x-4)Y-3 = 2(x-4)

Y-3 = 2x-8Y-3 = 2x-8 Y = 2x – 11 Y = 2x – 11 (slope-intercept form)(slope-intercept form)

Slope-Intercept ExplanationSlope-Intercept Explanation

y=mx+by=mx+b Sometimes in the Slope-intercept form, there are Sometimes in the Slope-intercept form, there are

fractions as the slope or the y-interceptfractions as the slope or the y-intercept B= y-interceptB= y-intercept Rise/RunRise/Run

When the slope is a fraction, you mark the B in a graph, When the slope is a fraction, you mark the B in a graph, which is the y-interceptwhich is the y-intercept

Then depending on the slope, if its positive than the line Then depending on the slope, if its positive than the line will look like this…will look like this…

If its not positive, but negative, it will look like: If its not positive, but negative, it will look like:

Point-Slope (Slope-Intercept) GraphPoint-Slope (Slope-Intercept) Graph

Y = 2x – 11Y = 2x – 11

(0,-11)(0,-11)

Rise/RunRise/RunGo up twice and to the side onceGo up twice and to the side once

(5,0)(5,0)

WebsitesWebsites(for further information)(for further information)

Linear EquationsLinear Equations http://www.algebralab.org/studyaids/studyaid.aspx?file=Algebra1_5-5.xmlhttp://www.algebralab.org/studyaids/studyaid.aspx?file=Algebra1_5-5.xml http://www.freemathhelp.com/point-slope.html http://www.freemathhelp.com/point-slope.html http://www.wonderhowto.com/how-to-solve-mixed-equation-decimal-percent-fraction-303082/ http://www.wonderhowto.com/how-to-solve-mixed-equation-decimal-percent-fraction-303082/


1- Properties1- Properties 2- Linear Equations2- Linear Equations

3- Linear Systems3- Linear Systems4- Solving 14- Solving 1stst Power Equations (1 Variable) Power Equations (1 Variable)5- Factoring5- Factoring6- Rational Expressions6- Rational Expressions7- Quadratic Equations7- Quadratic Equations8- Functions8- Functions9- Solving 19- Solving 1stst Power Inequalities (1 Variable) Power Inequalities (1 Variable)10- Word Problems10- Word Problems11- Extras11- Extras

Linear Systems— Method ExplanationLinear Systems— Method Explanation

SubstitutionSubstitution The Substitution Method, is used The Substitution Method, is used

when, there are two equations, and when, there are two equations, and you pick one (the one that looks the you pick one (the one that looks the easiest to do) and you isolate either easiest to do) and you isolate either the x or the ythe x or the y

When x or y is isolated, then you will When x or y is isolated, then you will get something like this:get something like this:

Y= ?x + ?Y= ?x + ? X= ?y + ?X= ?y + ?

Then, you replace the x or the y in the Then, you replace the x or the y in the equation that you didn’t touch yet, equation that you didn’t touch yet, and you must insert and you must insert

If you isolated the y, then you will If you isolated the y, then you will solve for xsolve for x

If you isolated the x, then you will If you isolated the x, then you will solve for ysolve for y

EliminationElimination The Elimination Method, is used The Elimination Method, is used

when there are two equations and, it when there are two equations and, it is said to be a lot easier than the is said to be a lot easier than the Substitution MethodSubstitution Method

First, you will have to decide whether First, you will have to decide whether you want to go for the x or the yyou want to go for the x or the y

Then, you will multiply and Then, you will multiply and cancel/eliminate either x or y cancel/eliminate either x or y depending, on which one did you depending, on which one did you chose to do (x or y)chose to do (x or y)

Then you solve for x or yThen you solve for x or y You will eventually substitute, more You will eventually substitute, more

like insert your y or x answer into the like insert your y or x answer into the either problem replacing it with x or y either problem replacing it with x or y

Then you solve for either x or yThen you solve for either x or y

Substitution MethodSubstitution Method

y = 11 - 4xy = 11 - 4x

x + 2(11 - 4x) = 8x + 2(11 - 4x) = 8

AnswersAnswers

Isolate the Y or XIsolate the Y or X

Substitute the number, insert itSubstitute the number, insert it

Solve for X and Solve for Y (vice versa)Solve for X and Solve for Y (vice versa)

Literal CoefficientsLiteral CoefficientsSimultaneous equations Simultaneous equations with literal coefficients with literal coefficients and literal constants and literal constants may be solved for the may be solved for the value of the variables value of the variables just as the other just as the other equations discussed in equations discussed in this chapter, with the this chapter, with the exception that the exception that the solution will contain solution will contain literal numbers. For literal numbers. For example, find the example, find the solution of the system:solution of the system:

We proceed as with any We proceed as with any other simultaneous linear other simultaneous linear equation. Using the equation. Using the addition method, we may addition method, we may proceed as follows: To proceed as follows: To eliminate the y term we eliminate the y term we multiply the first equation multiply the first equation by 3 and the second by 3 and the second equation by -4. The equation by -4. The equations then become …equations then become …

To eliminate x, we multiply To eliminate x, we multiply the first equation by 4 and the first equation by 4 and the second equation by -3. the second equation by -3. The equations then The equations then becomebecome

We may check in the same We may check in the same manner as that used for manner as that used for other equations, by other equations, by substituting these values substituting these values in the original equations.in the original equations.

http://www.tpub.com/math1/13d.htm

3 Variables !!3 Variables !!

Elimination MethodElimination Method 2x – 3y = 192x – 3y = 19 5x – 2y = 205x – 2y = 20

2x – 3y = 19 (2)2x – 3y = 19 (2) 5x – 2y = 20 (-3)5x – 2y = 20 (-3)

4x – 6y = 384x – 6y = 38 -15x + 6y = -60-15x + 6y = -60

-11x = -22-11x = -22 X = 2X = 2

2x – 3y = 192x – 3y = 19 2(2) – 3y = 192(2) – 3y = 19

4 – 3y = 194 – 3y = 19 -3y = 15-3y = 15

Y = -5Y = -5

The two equationsThe two equations

Now we multiply and then Now we multiply and then later cancel out a variable, later cancel out a variable, depending which one you depending which one you chosechose

Now we got one answer—x Now we got one answer—x = 2= 2

Now we must insert the two, Now we must insert the two, into the either of the into the either of the equationsequations…(substitution …(substitution method)method)

Now you got the y = -5Now you got the y = -5

DependentDependent

When a system is "dependent," it means that ALL When a system is "dependent," it means that ALL points that work in one of them ALSO work in the points that work in one of them ALSO work in the other one other one

Graphically, this means that one line is lying entirely Graphically, this means that one line is lying entirely on top of the other one, so that if you graphed both, on top of the other one, so that if you graphed both, you would really see only one line on the graph, since you would really see only one line on the graph, since they are imposed on top of each other they are imposed on top of each other

One of them totally DEPENDS on the other one One of them totally DEPENDS on the other one

IndependentIndependent

When a system is "independent," it means that they When a system is "independent," it means that they are not lying on top of each other are not lying on top of each other

There is EXACTLY ONE solution, and it is the point There is EXACTLY ONE solution, and it is the point of intersection of the two lines of intersection of the two lines

It's as if that one point is "independent" of the others. It's as if that one point is "independent" of the others. To sum up, a dependent system has INFINITELY To sum up, a dependent system has INFINITELY

MANY solutions. An independent system has MANY solutions. An independent system has EXACTLY ONE solutionEXACTLY ONE solution

ConsistentConsistent

We say that a point is a "solution" to the We say that a point is a "solution" to the system when it makes BOTH equations true, system when it makes BOTH equations true, right? right?

This is to say that there exists a point (or set of This is to say that there exists a point (or set of points) that "work" in one equation and also points) that "work" in one equation and also "work" in the other one "work" in the other one

So we say that this point is CONSISTENT So we say that this point is CONSISTENT from one equation to the next from one equation to the next

InconsistentInconsistent

On the other hand, if there are NO points that On the other hand, if there are NO points that work in both, then we say that the equations work in both, then we say that the equations are INCONSISTENT are INCONSISTENT

NO numbers that work in one are consistent NO numbers that work in one are consistent with the otherwith the other

To sum up, a consistent system has at least one To sum up, a consistent system has at least one solution. An inconsistent system has NO solution. An inconsistent system has NO solution at all solution at all

WebsitesWebsites

Linear Systems:Linear Systems: http://www.tpub.com/math1/13d.htmhttp://www.tpub.com/math1/13d.htm http://mathforum.org/library/drmath/view/62538.html http://mathforum.org/library/drmath/view/62538.html http://www.purplemath.com/modules/systlin2.htm http://www.purplemath.com/modules/systlin2.htm


1- Properties1- Properties 2- Linear Equations2- Linear Equations3- Linear Systems3- Linear Systems

4- Solving 14- Solving 1stst Power Equations (1 Variable) Power Equations (1 Variable)5- Factoring5- Factoring6- Rational Expressions6- Rational Expressions7- Quadratic Equations7- Quadratic Equations8- Functions8- Functions9- Solving 19- Solving 1stst Power Inequalities (1 Variable) Power Inequalities (1 Variable)10- Word Problems10- Word Problems11- Extras11- Extras

11stst Power Equations (1 Variable) Power Equations (1 Variable)

In order to get the answer, when there is only In order to get the answer, when there is only one variableone variable You must, isolate the variable, and if it has a sign You must, isolate the variable, and if it has a sign

with it (a negative sign) or a number with it, than with it (a negative sign) or a number with it, than you can and must divide the number to the other you can and must divide the number to the other sideside

In order to get the variable completely aloneIn order to get the variable completely alone Then you get your answerThen you get your answer

1 Variable Problems1 Variable Problems

5x + 3 = 2 (2 – 3x) 5x + 3 = 2 (2 – 3x) 5x + 3 = 4 – 6x5x + 3 = 4 – 6x 5x = 4 + (-3) – 6x5x = 4 + (-3) – 6x 5x = 1 – 6x5x = 1 – 6x 5x + 6x = 15x + 6x = 1 11x = 111x = 1 X = 1/11X = 1/11

2x = 82x = 8 X = 4X = 4

-x + 20 = – 3x + 2(5x – 10)-x + 20 = – 3x + 2(5x – 10) -x + 20 = – 3x + 10x – 20-x + 20 = – 3x + 10x – 20 -x = 7x – 40-x = 7x – 40 -8x = -40-8x = -40 X = 5X = 5

These 1 variable problems are These 1 variable problems are fairly simple and easyfairly simple and easy

All you have to do is isolate the All you have to do is isolate the variablevariable

Then just add, subtract, or divide Then just add, subtract, or divide and solve the problemand solve the problem


1- Properties1- Properties 2- Linear Equations2- Linear Equations3- Linear Systems3- Linear Systems4- Solving 14- Solving 1stst Power Equations (1 Variable) Power Equations (1 Variable)

5- Factoring5- Factoring6- Rational Expressions6- Rational Expressions7- Quadratic Equations7- Quadratic Equations8- Functions8- Functions9- Solving 19- Solving 1stst Power Inequalities (1 Variable) Power Inequalities (1 Variable)10- Word Problems10- Word Problems11- Extras11- Extras

Factoring FOILFactoring FOIL

FOILFOIL is a type of factoring that includes two “globs”is a type of factoring that includes two “globs”

(3x + 2)(3x – 2)(3x + 2)(3x – 2) this FOIL means that the O and I in FOIL, will be this FOIL means that the O and I in FOIL, will be

the same number, but one will be negative and one the same number, but one will be negative and one positive, therefore, they will cancel each other outpositive, therefore, they will cancel each other out

PSTPST

PST, is when two globs are reversed “FOILed” PST, is when two globs are reversed “FOILed” and they equal perfectlyand they equal perfectly

(x + 2)(x + 2)(x + 2)(x + 2) X + 4x + 4 (PST)X + 4x + 4 (PST)

The first number in a PST, to check if you got The first number in a PST, to check if you got a PST, the first number has to be squared, and a PST, the first number has to be squared, and if its not, then take out the GCFif its not, then take out the GCF

The First and Last number should have roots, The First and Last number should have roots, while the middle number should be the double while the middle number should be the double of the roots of both the First and Last numberof the roots of both the First and Last number

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Factor GCFFactor GCF

The GCF stands for the Greatest Common The GCF stands for the Greatest Common FactorFactor

Which means, that if you have a binomial or a Which means, that if you have a binomial or a trinomial with prime numbers in common or trinomial with prime numbers in common or more variables than needed, then you can more variables than needed, then you can factor them out, and then continue to solve the factor them out, and then continue to solve the problemproblem

Whatever you factored out, will still be part of Whatever you factored out, will still be part of the Answer of the problemthe Answer of the problem

Difference of SquaresDifference of Squares

First take out the GCF (always)First take out the GCF (always) If there are two globs that if FOILed, arent a PST, but If there are two globs that if FOILed, arent a PST, but

they just make a binomial, but it can be divided into they just make a binomial, but it can be divided into two more binomialstwo more binomials Then you have conjugatesThen you have conjugates (? + ?) (? - ?)(? + ?) (? - ?) As long as you have a negative glob, that you can still As long as you have a negative glob, that you can still

divide into more globs, you can continue to divide, but if divide into more globs, you can continue to divide, but if one glob is the same as another glob, then your answer will one glob is the same as another glob, then your answer will only contain the glob, but only onceonly contain the glob, but only once

Sum or Difference of CubesSum or Difference of Cubes

The Sum or Difference of Cubes, is when you The Sum or Difference of Cubes, is when you take variable squares or numbers with roots take variable squares or numbers with roots cubed, and they are separated and into a cubed, and they are separated and into a binomial and a trinomialbinomial and a trinomial

Reverse FOILReverse FOIL

This is the same thing as FOIL factoring, but This is the same thing as FOIL factoring, but there is a there is a Trial and Error systemTrial and Error system

That means, that when given trinomial, you That means, that when given trinomial, you will have to guess and check if it FOILs the will have to guess and check if it FOILs the correct globs, and you will have to continue to correct globs, and you will have to continue to do that, until you get the correct globsdo that, until you get the correct globs

Factor By GroupingFactor By Grouping

4x44x4 It is a binomial because there are two terms, and a It is a binomial because there are two terms, and a

repeated glob, it is a common globrepeated glob, it is a common glob Which means GCFWhich means GCF

2x22x2 Sometimes you can rearrange the order of the Sometimes you can rearrange the order of the

terms, to find the correct globterms, to find the correct glob

Factor By GroupingFactor By Grouping

3x13x1 Rearrange into a PSTRearrange into a PST Then make two perfect globsThen make two perfect globs If conjugates then separate themIf conjugates then separate them


1- Properties1- Properties 2- Linear Equations2- Linear Equations3- Linear Systems3- Linear Systems4- Solving 14- Solving 1stst Power Equations (1 Variable) Power Equations (1 Variable)5- Factoring5- Factoring

6- Rational Expressions6- Rational Expressions7- Quadratic Equations7- Quadratic Equations8- Functions8- Functions9- Solving 19- Solving 1stst Power Inequalities (1 Variable) Power Inequalities (1 Variable)10- Word Problems10- Word Problems11- Extras11- Extras

Rational ExpressionsRational Expressions

PSTPST X + 10x +25X + 10x +25

(x + 5)(x + 5)

We define a We define a Rational Rational ExpressionExpression as a fraction as a fraction where the numerator and the where the numerator and the denominator are denominator are polynomials in one or more polynomials in one or more

variables.variables.

22

22

3 5 8 (2)(4) 23 5 8 (2)(4) 2 + = = = + = = = 20 20 20 (4)(5) 5 20 20 20 (4)(5) 5

Addition and Subtraction of Addition and Subtraction of Rational ExpressionsRational Expressions

3x - 4x x(3x - 4) 3x - 43x - 4x x(3x - 4) 3x - 4 = = = = 2x - x x(2x - 1) 2x - 1 2x - x x(2x - 1) 2x - 1

22

22

Multiplication and Division of Multiplication and Division of Rational ExpressionsRational Expressions

R.ER.EFirst - multiply the first term in each set of parenthesis: 4x * x = 4x2

Outside - multiply the two terms on the outside: 4x * 2 = 8x

Inside - multiply both of the inside terms: 6 * x = 6x

Last - multiply the last term in each set of parenthesis: 6 x 2 = 12

FOILFOIL

WebsitesWebsites

Rational Expressions:Rational Expressions: http://www.freemathhelp.com/using-foil.html http://www.freemathhelp.com/using-foil.html


1- Properties1- Properties 2- Linear Equations2- Linear Equations3- Linear Systems3- Linear Systems4- Solving 14- Solving 1stst Power Equations (1 Variable) Power Equations (1 Variable)5- Factoring5- Factoring6- Rational Expressions6- Rational Expressions

7- Quadratic Equations7- Quadratic Equations8- Functions8- Functions9- Solving 19- Solving 1stst Power Inequalities (1 Variable) Power Inequalities (1 Variable)10- Word Problems10- Word Problems11- Extras11- Extras

Completing the Square (1)Completing the Square (1) (X + n)² = X² + 2nx + n²(X + n)² = X² + 2nx + n² Note the rightmost term (n²) is relatedNote the rightmost term (n²) is related

to 2n (the x coefficient) by the formulato 2n (the x coefficient) by the formula Solving this by "Solving this by "completing the squarecompleting the square" is as follows:" is as follows:

1) Move the "non X" term to the right: 1) Move the "non X" term to the right:

4X² + 12X = 164X² + 12X = 16

2) Divide the equation by the coefficient of X² which in this case is 4 2) Divide the equation by the coefficient of X² which in this case is 4 X² + 3X = 4X² + 3X = 4 3) Now here's the "3) Now here's the "completing the squarecompleting the square" stage in which we: " stage in which we: • • take the coefficient of Xtake the coefficient of X

• divide it by 2 • divide it by 2 • square that number • square that number • then add it to both sides of the • then add it to both sides of the equation. equation.

Completing the Square (2)Completing the Square (2) In our sample problem the coefficient of X is 3In our sample problem the coefficient of X is 3

dividing this by 2 equals 1.5 dividing this by 2 equals 1.5 squaring this number equals (1.5)² = 2.25 squaring this number equals (1.5)² = 2.25 Now, adding that to both sides of the equation, we have: Now, adding that to both sides of the equation, we have:

X² + 3X + 2.25 = 4 + 2.25X² + 3X + 2.25 = 4 + 2.25 4) 4) FinallyFinally, we can take the square root of , we can take the square root of both sidesboth sides of the equation and we have: of the equation and we have:

X + 1.5 = Square Root (4 + 2.25)X + 1.5 = Square Root (4 + 2.25)

X = Square Root (6.25) -1.5X = Square Root (6.25) -1.5

X = 2.5 -1.5 X = 2.5 -1.5

X = 1.0X = 1.0 Let's not forget that the Let's not forget that the otherother square root of 6.25 is -2.5 and so the other root of the equation square root of 6.25 is -2.5 and so the other root of the equation

is:is:

(-2.5 -1.5) = -4(-2.5 -1.5) = -4

Quadratic FormulaQuadratic Formula

We can follow precisely the same procedure as above to derive the Quadratic Formula. All Quadratic Equations have the general form:

aX² + bX + c = 0

Discriminant and the Quadratic Discriminant and the Quadratic EquationEquation

The Discriminant is a number that can be calculated The Discriminant is a number that can be calculated from any quadratic equation A quadratic equation is from any quadratic equation A quadratic equation is an equation that can be written as an equation that can be written as

ax ² + bx + c where a ≠ 0ax ² + bx + c where a ≠ 0 The Discriminant in a quadratic equation is found by The Discriminant in a quadratic equation is found by

the following formula and the discriminant provides the following formula and the discriminant provides critical information regarding the nature of the critical information regarding the nature of the roots/solutions of any quadratic equation. roots/solutions of any quadratic equation. discriminant= b² − 4acdiscriminant= b² − 4acExample of the discriminant Example of the discriminant

Quadratic equation = y = 3x² + 9x + 5 Quadratic equation = y = 3x² + 9x + 5 The discriminant = 9 ² − 4 • 3 •5 The discriminant = 9 ² − 4 • 3 •5

Quadratic EquationQuadratic EquationQuadratic Equation: y = x² + 2x + 1y = x² + 2x + 1

•a = 1

•b = 2 •c = 1 •The discriminant for this equation is 2² - 4•1 •1= 4 − 4 = 0 Since the discriminant of zero, there should be 1 real solution to this equation. Below is a picture representing the graph and one solution of this quadratic equation Graph of y = x² + 2x +1

WebsitesWebsites

R.E.:R.E.: http://www.mathwarehouse.com/quadratic/discriminant-in-quadratic-equation.php http://www.mathwarehouse.com/quadratic/discriminant-in-quadratic-equation.php http://webgraphing.com/quadraticequation_quadraticformula.jsp http://webgraphing.com/quadraticequation_quadraticformula.jsp http://webmath.com/quadtri.html http://webmath.com/quadtri.html


1- Properties1- Properties 2- Linear Equations2- Linear Equations3- Linear Systems3- Linear Systems4- Solving 14- Solving 1stst Power Equations (1 Variable) Power Equations (1 Variable)5- Factoring5- Factoring6- Rational Expressions6- Rational Expressions7- Quadratic Equations7- Quadratic Equations

8- Functions8- Functions9- Solving 19- Solving 1stst Power Inequalities (1 Variable) Power Inequalities (1 Variable)10- Word Problems10- Word Problems11- Extras11- Extras

F(x)F(x)

In Algebra f(x) is another symbol for yIn Algebra f(x) is another symbol for y Y = 3Y = 3 F(x) = 3F(x) = 3

Its practically the same things, but people use it for Its practically the same things, but people use it for confusion confusion

Domain and RangeDomain and Range

DomainDomain For a function f defined by an expression with For a function f defined by an expression with

variable x, the implied domain of f is the set of all variable x, the implied domain of f is the set of all real numbers variable x can take such that the real numbers variable x can take such that the expression defining the function is real. The expression defining the function is real. The domain can also be given explicitly. domain can also be given explicitly.

RangeRange The range of f is the set of all values that the The range of f is the set of all values that the

function takes when x takes values in the domain function takes when x takes values in the domain

DomainDomainExample: The function y = √(x + 4) has the following graph

•The domain of the function is x ≥ −4, since x cannot take values less than −4. (Try some values in your calculator, some less than −4 and some more than −4. The only ones that "work" and give us an answer are the ones greater than or equal to −4).•Note:

•The enclosed (colored-in) circle on the point (-4, 0). This indicates that the domain "starts" at this point. •That x can take any positive value in this example

RangeRangeExample 1: Let's return to the example above, y = √(x + 4). We notice that there are only positive y-values. There is no value of x that we can find such that we will get a negative value of y. We say that the range for this function is y ≥ 0.

Example 2: The curve of y = sin x shows the range to be betweeen −1 and 1

The domain of the function y = sin x is "all values of x", since there are no restrictions on the values for x.

http://www.intmath.com/Functions-and-http://www.intmath.com/Functions-and-graphs/2a_Domain-and-range.php graphs/2a_Domain-and-range.php


1- Properties1- Properties 2- Linear Equations2- Linear Equations3- Linear Systems3- Linear Systems4- Solving 14- Solving 1stst Power Equations (1 Variable) Power Equations (1 Variable)5- Factoring5- Factoring6- Rational Expressions6- Rational Expressions7- Quadratic Equations7- Quadratic Equations8- Functions8- Functions

9- Solving 19- Solving 1stst Power Inequalities (1 Variable) Power Inequalities (1 Variable)10- Word Problems10- Word Problems11- Extras11- Extras

Solving InequalitiesSolving Inequalities Linear inequalities Linear inequalities are also called first degree inequalities, as are also called first degree inequalities, as

the highest power of the variable in these inequalities is 1. the highest power of the variable in these inequalities is 1. E.g. 4E.g. 4x x > 20 is an inequality of the first degree, which is often > 20 is an inequality of the first degree, which is often called a linear inequality.called a linear inequality.

Many problems can be solved using linear inequalities.Many problems can be solved using linear inequalities. We know that a linear equation with one pronumeral has only We know that a linear equation with one pronumeral has only

one value for the solution that holds true. For example, the one value for the solution that holds true. For example, the linear equation 6linear equation 6xx = = 24 is a true statement only when 24 is a true statement only when xx = 4. = 4. However, the linear inequality 6However, the linear inequality 6x > x > 24 is satisfied when 24 is satisfied when x > x > 4. 4. So, there are many values of So, there are many values of xx which will satisfy the inequality which will satisfy the inequality 66x > x > 24.24.

InequalitiesInequalities

Recall that:Recall that: the same number can be subtracted from both sides of an the same number can be subtracted from both sides of an

inequality inequality the same number can be added to both sides of an inequality the same number can be added to both sides of an inequality both sides of an inequality can be multiplied (or divided) by both sides of an inequality can be multiplied (or divided) by

the same positive number the same positive number if an inequality is multiplied (or divided) by the same negative if an inequality is multiplied (or divided) by the same negative

number, then:number, then:

InequalitiesInequalities

ConjunctionsConjunctions

When two inequalities are joined by the word When two inequalities are joined by the word andand or the word or the word oror, a , a compound inequalitycompound inequality is is formed. A compound inequality likeformed. A compound inequality like-3 < 2x + 5 -3 < 2x + 5 andand 2x + 5 ≤ 7 2x + 5 ≤ 7is called a is called a conjunctionconjunction, because it uses the , because it uses the word word andand. The sentence -3 < 2x + 5 ≤ 7 is an . The sentence -3 < 2x + 5 ≤ 7 is an abbreviation for the preceding conjunction. abbreviation for the preceding conjunction. Compound inequalities can be solved using the Compound inequalities can be solved using the addition and multiplication principles for addition and multiplication principles for inequalities. inequalities.

DisjunctionDisjunction

A compound inequality like 2x - 5 ≤ -7 A compound inequality like 2x - 5 ≤ -7 oror is is called a called a disjunctiondisjunction, because it contains the , because it contains the word word oror. Unlike some conjunctions, it cannot . Unlike some conjunctions, it cannot be abbreviated; that is, it cannot be written be abbreviated; that is, it cannot be written without the word without the word oror..


1- Properties1- Properties 2- Linear Equations2- Linear Equations3- Linear Systems3- Linear Systems4- Solving 14- Solving 1stst Power Equations (1 Variable) Power Equations (1 Variable)5- Factoring5- Factoring6- Rational Expressions6- Rational Expressions7- Quadratic Equations7- Quadratic Equations8- Functions8- Functions9- Solving 19- Solving 1stst Power Inequalities (1 Variable) Power Inequalities (1 Variable)

10- Word Problems10- Word Problems11- Extras11- Extras

Word ProblemsWord Problems1)1) The sum of twice a number plus 13 is 75. Find the number.The sum of twice a number plus 13 is 75. Find the number. The word The word isis means means equalsequals. The word . The word andand means means plusplus. .

Therefore, you can rewrite the problem like the following: Therefore, you can rewrite the problem like the following: The sum of twice a number and 13 equals 75The sum of twice a number and 13 equals 75 . .

Using numbers and a variable that represents something, Using numbers and a variable that represents something, NN in this in this case (for case (for numbernumber), you can write an equation that means the same ), you can write an equation that means the same thing as the original problem. thing as the original problem. 22NN + 13 = 75 + 13 = 75

Solve this equation by isolating the variable. Solve this equation by isolating the variable. 22NN + 13 = 75 Equation. - 13 = -13 Add (-13) to both sides. + 13 = 75 Equation. - 13 = -13 Add (-13) to both sides. ------------- 2------------- 2NN = 62 = 62

NN = 31 = 31 Divided both sides by 2 Divided both sides by 2

Word ProblemsWord Problems

2) 2) Find a number which decreased by 18 is 5 Find a number which decreased by 18 is 5 times its opposite.times its opposite.

Again, you look for words that describe equal quantities. Is means equals, and decreased by means minus. Also, opposite always means negative. Keeping that information in mind makes it so an equation can be written that describes the problem, just like the following: N - 18 = 5(-N) Equation. N - 18 = -5N Multiplied out. 5N + 18 5N + 18 Add (5N + 18) to ------------------ both sides. 6N = 18 N = 3 Divide both sides by 6 to isolate N.


3) 3) Julie has $50, which is eight dollars more Julie has $50, which is eight dollars more than twice what John has. How much has than twice what John has. How much has John? First, what will you let John? First, what will you let xx represent? represent?

The unknown number -- which is how much The unknown number -- which is how much that John has.that John has.

What is the equation?What is the equation? 22xx + 8 = 50. + 8 = 50. Here is the solution:Here is the solution: xx = $21 = $21


4) 4) Carlotta spent $35 at the market. This was Carlotta spent $35 at the market. This was seven dollars less than three times what she seven dollars less than three times what she spent at the bookstore; how much did she spent at the bookstore; how much did she spend there?spend there?

Here is the equation. Here is the equation. 33xx − 7 = 35 − 7 = 35 Here is the solution:Here is the solution: xx = $14 = $14


1- Properties1- Properties 2- Linear Equations2- Linear Equations3- Linear Systems3- Linear Systems4- Solving 14- Solving 1stst Power Equations (1 Variable) Power Equations (1 Variable)5- Factoring5- Factoring6- Rational Expressions6- Rational Expressions7- Quadratic Equations7- Quadratic Equations8- Functions8- Functions9- Solving 19- Solving 1stst Power Inequalities (1 Variable) Power Inequalities (1 Variable)10- Word Problems10- Word Problems11- Extras11- Extras

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