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Algebra 1

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Algebra 1

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  1. Algebra 1 Marcos De la Cruz Algebra 1(6th period) Ms.Hardtke 5/14/10

  2. Algebra Topics 1- Properties 2- Linear Equations 3- Linear Systems 4- Solving 1st Power Equations (1 Variable) 5- Factoring 6- Rational Expressions 7- Quadratic Equations 8- Functions 9- Solving 1st Power Inequalities (1 Variable) 10- Word Problems 11- Extras

  3. If the same number is added to both sides of an equation, both sides will be and remain equal 3=3 (equation) If 2 is added to both sides 2+3=3+2 5=5 Negative Special Case y=3x+5 (equation) If (-3) is added to both sides Y-3=3x+5-3 Y-3=3x+2 Its still equal Addition Property of Equality

  4. Multiplication Property of Equality • States that when both sides of an equal equation is multiplied and the equation remains equal • If 5=5 (equation) • 5x2=2x5 • You multiply both sides by 2 • 10=10 • Still remains equal

  5. Reflexive Property of Equality • When something is the exact same on both sides • A = A • 7x = 7x • 3456x = 3456x

  6. Symmetric Property of Equality • When two variables are different but are the same number/amount (equal symmetry) • If a=b, then b=a • If c=d, then d=c • If xyp=xyo, then xyo=xyp

  7. Transitive Property of Equality • When numbers or variables are all equal • If a=b and b=c, then c=a • if 5x=100 and 100=4y, then 4y=5x • if 0=2x and 2x=78p, then 78p=0

  8. Associative Property of Addition • The sum of a set of numbers is the same no matter how the numbers are grouped. Associative property of addition can be summarized algebraically as: (a + b) + c = a + (b + c) • (3 + 5) + 2 = 8 + 2 = 10 • 3 + (5 + 2) = 3 + 7 = 10 • (3 + 5) + 2 = 3 + (5 + 2).

  9. Associative Property of Multiplication • The product of a set of numbers is the same no matter how the numbers are grouped. The associative property of multiplication can be summarized algebraically as: (ab)c = a(bc)

  10. Commutative Property of Addition • The sum of a group of numbers is the same regardless of the order in which the numbers are arranged. Algebraically, the commutative property of addition states: a + b = b + a • 5 + 2 = 2 + 5 because 5 + 2 = 7 and 2 + 5 = 7 • -3 + 11 = 11 - 3

  11. Commutative Property of Multiplication • The product of a group of numbers is the same regardless of the order in which the numbers are arranged. Algebraically, commutative property of multiplication can be written as: ab = ba • 8x6 = 48 and 6x8 = 48: thus, 8x6 = 6x8

  12. Distributive Property • The sum of two addends multiplied by a number is the sum of the product of each addend and the number • A(b+c) • Ab + Ac • 3x(2y+4) • 6xy + 12x

  13. Property of Opposites/Inverse Property of Addition • When a number is added by itself negative or positive to make zero a + (-a) = 0 • 5 + (-5) = 0 • -3y + (3y) = 0

  14. For two ratios, if a/b = c/d, then b/a = d/c a(1/a) = 1 5(1/5) = 1 8/1 x 1/8 = 1 A number times its reciprocal, always equals one A Reciprocal is its reverse and opposite (the signs switch from + to — or vice versa) Property Of Reciprocals/Inverse Property of Multiplication

  15. Reciprocal Function(continued) • The reciprocal function: y = 1⁄x. For every x except 0, y represents its multiplicative inverse

  16. Identity Property of Addition • A number that can be added to any second number without changing the second number. Identity for addition is 0 (zero) since adding zero to any number will give the number itself: 0 + a = a + 0 = a • 0 + (-3) = (-3) + 0 = -3 • 0 + 5 = 5 + 0 = 5

  17. Identity Property of Multiplication • A number that can be multiplied by any second number without changing the second number. Identity for multiplication is "1,“ instead of 0, because multiplying any number by 1 will not change it. a x 1 = 1 x a = a • (-3) x 1 = 1 x (-3) = -3 • 1 x 5 = 5 x 1 = 5

  18. Multiplicative Property of Zero • Anything number or variable multiplied times zero (0), will always equal zero • 5 x 0=0 • 5g x 0=0 No matter what number is being multiplied by zero, it will always be zero A really long way to explain the Multiplicative Property of Zero (Proof) http://upload.wikimedia.org/math/b/5/8/b5892630f1d2f28a580331a1d7e3e79f.png

  19. Closure Property of Addition • Sum (or difference) of 2 real numbers equals a real number • 10 – (5)= 5

  20. Closure Property of Multiplication • Product (or quotient if denominator 0) of 2 Reals equals a real number • 5 x 2 = 10

  21. Exponents Exponents are the little numbers above numbers, that mean that the number is multiplied by itself that many times 7 × 7 = (7 × 7) × (7 × 7 × 7 × 7 × 7 × 7) When two exponents or numbers with exponents are being multiplied, you add both exponents, but you still multiply the number or variable 3x (5x ) = 15x 15x (Exponents) Product of Powers Property 3 4 2 6 (3+4) 7

  22. Power of a Product Property • To find a power of a product, find the power of each factor and then multiply.  In general: (ab) = a · b Or  a · b = (ab) • (3t) • (3t) = 3 · t = 81t m m m m m m 4 4 4 4 4

  23. Power of a Power Property • To find a power of a power, multiply the exponents. (Its basically the same as the Power of a Product Property, if forgotten, go one slide back and review.) • (5 ) • (5 )(5 )(5 )(5 ) = 5 5 • Its basically this: (a ) = a 3 4 3 3 3 3 3(4) 12 b c bc

  24. When both the denominator and numerator of a fraction have a common variable, it can be canceled, therefore not usable anymore Also when a variable is canceled, the exponents are subtracted, instead of added as in the Product of Powers Property a /a a 5 5x5x5 —— —————— 5 5x5 = 5 (the canceling of common factors) Quotient of Powers Property b c b-c 3 2

  25. Power of a Quotient Property • This is almost the same as the Quotient of Powers Property, but this time, an entire fraction is multiplied by an exponent • You also have to cancel the common factors, if there are any • (a/b) a /b — (a/6) (and vice versa) (a /36) c c c 2 2

  26. Zero Power Property • If a variable has an exponent of zero, then it must equal one • a =1 • b =1 • c b a =1 • (a ) =1 0 0 0 0 0 2 0

  27. Negative Power Property • When a fraction or a number has negative exponents, you must change it to its reciprocal in order to turn the negative exponent into a positive exponent • 4 ¼ 1/16 -2 2 The exponent turned from negative to positive

  28. Zero Product Property • When both variables equal zero, then one or the other must equal zero • if ab=0, then either a=0 or b=0 • if xy=0, then either x=0 or y=0 • if abc=0, then either a=0, b=0, or c=0

  29. Product of Roots Property • The product is the same as the product of square roots

  30. Quotient of Roots Property • The square root of the quotient is the same as the quotient of the square roots: A A B B

  31. Root of a Power Property

  32. Power of a Root Property

  33. Density Property of Rational Numbers • Between any two rational numbers, there exists at least one additional rational number 1 2 3 4 5 6 7 8 9 4.5 or 4½

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

  35. Algebra Topics 1- Properties 2- Linear Equations 3- Linear Systems 4- Solving 1st Power Equations (1 Variable) 5- Factoring 6- Rational Expressions 7- Quadratic Equations 8- Functions 9- Solving 1st Power Inequalities (1 Variable) 10- Word Problems 11- Extras

  36. Standard Form Ax + By = C The terms A, B, and C are integers (could be either positive or negative numbers or fractions) If Fractions: Multiply each term in the equation by its LCD (Lowest Common Denominator) Either add or subtract to get either X or Y isolated, in one side of the = If Decimals: Multiply each term in the equation depending on the decimal with the most numbers (by 10, 100, 1000, etc) 1.23 (multiply times 100) 123.00 Subtract or add to get X or Y isolated If Normal Numbers (neither fractions or decimals): Just add or subtract to get X or Y isolated Standard/General Form

  37. Graph Points • A Graph Point contains of an X and a Y (x,y) • The X and Y mean where exactly the point is located Y line graph X line graph

  38. Fractions: You multiply by the LCM Which in this case is 20x Then to double check it… Standard/General Form Ex.

  39. The Point-Slope form. got its name because it uses a single point in a graph and a on the slope of the line It is usually used to find the slope of a graph, if the slope is not given in a certain problem or equation The Y on the Point-Slope form., doesn’t mean that the Y is multiplied by one, but it means to use the first Y of the two or one point given as a problem (same with X) (4,3) and the slope is 2 M = slope Y—stays the same X —is 4 (because 4 is in the x spot) Y — is 3 X—stays the same If the problem gives you two points and no slope, then you are free to choose what which or the Xs or the Ys you may want to use for your Point-Slope Form. Point-Slope Form 1 ex 1 1

  40. Point-Slope Form • (4,3) and m=2 • you must convert “it” to a slope-intercept form Y=Mx + B • Y-3 = 2(x-4) • Y-3 = 2x-8 • Y = 2x – 11 (slope-intercept form)

  41. Slope-Intercept Explanation • y=mx+b • Sometimes in the Slope-intercept form, there are fractions as the slope or the y-intercept • B= y-intercept • Rise/Run • When the slope is a fraction, you mark the B in a graph, which is the y-intercept • Then depending on the slope, if its positive than the line will look like this… • If its not positive, but negative, it will look like:

  42. Point-Slope (Slope-Intercept) Graph • Y = 2x – 11 Rise/Run Go up twice and to the side once (5,0) (0,-11)

  43. Websites(for further information) Linear Equations • http://www.algebralab.org/studyaids/studyaid.aspx?file=Algebra1_5-5.xml • http://www.freemathhelp.com/point-slope.html • http://www.wonderhowto.com/how-to-solve-mixed-equation-decimal-percent-fraction-303082/

  44. Algebra Topics 1- Properties 2- Linear Equations 3- Linear Systems 4- Solving 1st Power Equations (1 Variable) 5- Factoring 6- Rational Expressions 7- Quadratic Equations 8- Functions 9- Solving 1st Power Inequalities (1 Variable) 10- Word Problems 11- Extras

  45. Substitution The Substitution Method, is used when, there are two equations, and you pick one (the one that looks the easiest to do) and you isolate either the x or the y When x or y is isolated, then you will get something like this: Y= ?x + ? X= ?y + ? Then, you replace the x or the y in the equation that you didn’t touch yet, and you must insert If you isolated the y, then you will solve for x If you isolated the x, then you will solve for y Elimination The Elimination Method, is used when there are two equations and, it is said to be a lot easier than the Substitution Method First, you will have to decide whether you want to go for the x or the y Then, you will multiply and cancel/eliminate either x or y depending, on which one did you chose to do (x or y) Then you solve for x or y You will eventually substitute, more like insert your y or x answer into the either problem replacing it with x or y Then you solve for either x or y Linear Systems— Method Explanation

  46. Substitution Method y = 11 - 4x Isolate the Y or X Substitute the number, insert it x + 2(11 - 4x) = 8 Solve for X and Solve for Y (vice versa) Answers

  47. Literal Coefficients http://www.tpub.com/math1/13d.htm Simultaneous equations with literal coefficients and literal constants may be solved for the value of the variables just as the other equations discussed in this chapter, with the exception that the solution will contain literal numbers. For example, find the solution of the system: We proceed as with any other simultaneous linear equation. Using the addition method, we may proceed as follows: To eliminate the y term we multiply the first equation by 3 and the second equation by -4. The equations then become … To eliminate x, we multiply the first equation by 4 and the second equation by -3. The equations then become We may check in the same manner as that used for other equations, by substituting these values in the original equations. 3 Variables !!

  48. 2x – 3y = 19 5x – 2y = 20 2x – 3y = 19 (2) 5x – 2y = 20 (-3) 4x – 6y = 38 -15x + 6y = -60 -11x = -22 X = 2 2x – 3y = 19 2(2) – 3y = 19 4 – 3y = 19 -3y = 15 Y = -5 The two equations Now we multiply and then later cancel out a variable, depending which one you chose Now we got one answer—x = 2 Now we must insert the two, into the either of the equations…(substitution method) Now you got the y = -5 Elimination Method

  49. Dependent • When a system is "dependent," it means that ALL points that work in one of them ALSO work in the other one • Graphically, this means that one line is lying entirely on top of the other one, so that if you graphed both, you would really see only one line on the graph, since they are imposed on top of each other • One of them totally DEPENDS on the other one

  50. Independent • When a system is "independent," it means that they are not lying on top of each other • There is EXACTLY ONE solution, and it is the point of intersection of the two lines • It's as if that one point is "independent" of the others. • To sum up, a dependent system has INFINITELY MANY solutions. An independent system has EXACTLY ONE solution