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By: Max Kent

Algebra 1. By: Max Kent. For Help refer to :. http://www.mathtv.com/. http://www.quickmath.com/. Addition Property (of Equality). If the same number is added to both sides of an equation, the two sides remain equal. That is if x = y , then x + z = y + z.

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By: Max Kent

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  1. Algebra 1 By: Max Kent For Help refer to : http://www.mathtv.com/ http://www.quickmath.com/

  2. Addition Property (of Equality) If the same number is added to both sides of an equation, the two sides remain equal. That is if x = y, then x + z = y + z. Multiplication Property (of Equality) For all real numbers  a  and  b , and for  c ≠ 0 ,a = b     is equivalent to     ac = bc .

  3. Reflexive Property (of Equality) The property that a = a. Symmetric Property (of Equality) The following property: If if a = b then b = a. Transitive Property (of Equality) The following property: If a = b and b = c, then a = c. One of the equivalence properties of equality. Note: This is a property of equality and inequalities. One must be cautious, however, when attempting to develop arguments using the transitive property in other settings. Here is an example of an unsound application of the transitive property: "Team A defeated team B, and team B defeated team C. Therefore, team A will defeat team C.“ http://www.mathwords.com/t/transitive_property.htm

  4. Associative Property of Addition The addition or multiplication of a set of numbers is the same regardless of how the numbers are grouped. The associative property will involve 3 or more numbers. The parenthesis indicates the terms that are considered one unit. The groupings (Associative Property) are within the parenthesis. Hence, the numbers are 'associated' together. In multiplication, the product is always the same regardless of their grouping. The Associative Property is pretty basic to computational strategies. Remember, the groupings in the brackets are always done first, this is part of the order of operations. When we change the groupings of addends, the sum does not change:(2 + 5) + 4 = 11 or 2 + (5 + 4) = 11 Associative Property of Multiplication When three or more numbers are multiplied, the product is the same regardless of the grouping of the factors. Example: (2 * 3) * 4 = 2 * (3 * 4)

  5. Commutative Property of Addition The Commutative Property of Addition states that changing the order of addends does not change the sum, i.e. if a and b are two real numbers, then a + b = b + a. Commutative Property of Multiplication The commutative property of multiplication simply means it does not matter which number is first when you write the problem.  The answer is the same.3 x 5 = 5 x 3  (The numbers can be switched around and the answer is the same.)    

  6. Distributive Property (of Multiplication over Addition) The distributive property of multiplication over addition is simply this:  it makes no difference whether you add two or more terms together first, and then multiply the results by a factor, or whether you multiply each term alone by the factor first, and then add up the results. That is, adding up the term first; then multiplying by the factor   =  multiplying each term by the factor first, then adding up the resulting terms That is:       Factor(Term1 + Term2 + ... + TermN)  =   Factor(Term1) + Factor(Term2) + ..... + Factor(TermN) If we call the Factor "a,"  and we call the terms "b", "c,"......"t", then this statement begins to look like a mathematical statement: a(b + c + ....... + t)    = a(b) + a(c) + .... +a(t) EXAMPLE:    (The factor is 3, and the three terms  are 2, 7, -5) 3(2 + 7 - 5)  =   3(2) + 3(7) + (3)(-5) 3(4)         =     6    +  21    -  15 12         =   12

  7. Prop of Opposites or Inverse Property of Addition When you add a number to its opposite you get zero a+(-a)=0 Prop of Reciprocals or Inverse Prop. of Multiplication A reciprocal is the number you have to multiply a given number by to get 1. Ex) you have to multiply 2 by 1/2 to get 1. therefore the reciprocal of 2 is 1/2 As implied above, a property of two reciprocals is that their product equals 1. Another name for "reciprocal" is "multiplicative inverse."

  8. Identity Property of Addition Identity property of addition states that the sum of zero and any number or variable is the number or variable itself. For example, 4 + 0 = 4, - 11 + 0 = - 11, y + 0 = y are few examples illustrating the identity property of addition. Identity Property of Multiplication Identity property of multiplication states that the product of 1 and any number or variable is the number or variable itself. For example, 4 × 1 = 4, - 11 × 1 = - 11, y × 1 = y are few examples illustrating the identity property of multiplication.

  9. Multiplicative Property of Zero A number times zero equals zero. (7*0=0) Closure Property of Addition The closure property of addition says that if you add together any two numbers from a set, you will get another number from the same set. If the sum is not a number in the set, then the set is not closed under addition. Closure Property of Multiplication Closure Property:For any two whole numbers a and b, their product a*b is also a whole number.Example: 10*9 = 90

  10. Product of Powers Property How do you simplify 72 × 76? If you recall the way exponents are defined, you know that this means: (7 × 7) × (7 × 7 × 7 × 7 × 7 × 7) If we remove the parentheses, we have the product of eight 7s, which can be written more simply as: 78 This suggests a shortcut: all we need to do is add the exponents! 72 × 76 = 7(2 + 6) = 78 In general, for all real numbers a, b, and c, ab × ac = a(b + c) To multiply two powers with the same base, add the exponents. Power of a Product Property To find a power of a product, find the power of each factor and then multiply.  In general, (ab)m = am · bm. Power of a Power Property To find a power of a power, multiply the exponents.  This is an extension of the product of powers property. Suppose you have a number raised to a power, and you multiply the whole expression by itself over and over. This is the same as raising the expression to a power: (53)4 = (53)(53)(53)(53)

  11. Quotient of Powers Property This property states that to divide powers having the same base, subtract the exponents. Power of a Quotient Property This property states that the power of a quotient can be obtained by finding the powers of numerator and denominator and dividing them.

  12. Zero Power Property A number to the power of zero equals 1. 2130457040=1 Negative Power Property When you have a negative exponent on, say, 4, it will be written 4^-2 You basically take the reciprocal of it and change the exponent to a positive one. 4^-2 would be 1/4^2

  13. Zero Product Property The Zero Product Property simply states that if ab = 0, then either a = 0 or b = 0 (or both). A product of factors is zero if and only if one or more of the factors is zero.

  14. Product of Roots Property The product of the square roots is the square root of the product. Quotient of Roots Property For any non-negative (positive or 0) real number a and any positive real number b: =√a -- √b

  15. Root of a Power Property Power of a Root Property

  16. Quiz Which property? X+Y=Y+X Click for answer Commutative Property (of Addition)

  17. Quiz Which property? (5x+9x)+3x=(5x+3x)+9X Click for answer Associative Property of Addition

  18. Solving 1st power inequalities in one variable. • With only one inequality sign = 3x<15 | x<5 • Conjunction = 3x<15<5x | x<5 and x>3 • Disjunction = 2x>8 or 2x<4 | x>4 or x<2 Solving Inequalities Linear inequalities are also called first degree inequalities, as the highest power of the variable (or pronumeral) in these inequalities is 1. E.g.  4x > 20 is an inequality of the first degree, which is often called a linear inequality. Many problems can be solved using linear inequalities.We know that a linear equation with one pronumeral has only one value for the solution that holds true. For example, the linear equation 6x= 24 is a true statement only when x = 4. However, the linear inequality 6x > 24 is satisfied when x > 4. So, there are many values of x which will satisfy the inequality 6x > 24. Similarly, we can show that all numbers greater than 4 satisfy this inequality.

  19. Linear equations in two variables Graph: y=x-5?

  20. Linear Systems Solve: y = 3x – 2y = –x – 6 Y=-x-6 3x-2=-x-6 4x=8 X=2 Now solve for Y Y=3(2)-2 Y=6-2 Y=4 The answer is (2,4)

  21. Factoring Solve: X2+10x+25+y2? [X2+10x+25] is a PST (x+5)2 +y2 Now just factor y and put it in with each binomial The answer will be: (x+5+y)(x+5+y)

  22. Rational Expressions Solve:

  23. Functions f(x)= is another way to write y= Functions are relations only when every input has a distinct output, so not all relations are functions but all functions are relations. Let’s say you had the points (2,3) and (3,4) and you needed to find a linear function that contained them. This is how you would do that. 3-4 over (divided by) 2-3 (rise over run, Y is rise, X is run) you would get -1 over -1. This equals 1, which will be the slope. To find y-intercept, substitute: 2=1(3)+b 2=3+b  -1=b So your final equation is: Y=X-1. You can now graph this.

  24. Parabolas continued Graph: x2-6x+5 The x-intercepts are (5,0) and (1,0).  y-intercept: The y-intercept is (0,5). and Vertex: So the vertex is (3, -4).

  25. Simplifying Expressions With Exponents The "minus" on the 2 says to move the variable; the "minus" on the 6 says that the 6 is negative. Warning: These two "minus" signs mean entirely different things, and should not be confused. I have to move the variable; I should not move the 6. Simplify: Answer:

  26. Simplifying expressions with radicals Example:

  27. Word problems Solve: You need a 15% acid solution for a certain test, but your supplier only ships a 10% solution and a 30% solution. Rather than pay the hefty surcharge to have the supplier make a 15% solution, you decide to mix 10% solution with 30% solution, to make your own 15% solution. You need 10 liters of the 15% acid solution. How many liters of 10% solution and 30% solution should you use? Let x stand for the number of liters of 10% solution, and let y stand for the number of liters of 30% solution. (The labeling of variables is, in this case, very important, because "x" and "y" are not at all suggestive of what they stand for. If we don't label, we won't be able to interpret our answer in the end.) For mixture problems, it is often very helpful to do a grid:

  28. More Word Problems Solve: A collection of 33 coins, consisting of nickels, dimes, and quarters, has a value of $3.30. If there are three times as many nickels as quarters, and one-half as many dimes as nickels, how many coins of each kind are there? I'll start by picking and defining a variable, and then I'll use translation to convert this exercise into mathematical expressions. Nickels are defined in terms of quarters, and dimes are defined in terms of nickels, so I'll pick a variable to stand for the number of quarters, and then work from there: number of quarters: q number of nickels: 3q number of dimes: (½)(3q) = (3/2)q There is a total of 33 coins, so: q + 3q + (3/2)q = 33 4q + (3/2)q = 33 8q + 3q = 66 11q = 66 q = 6 Then there are six quarters, and I can work backwards to figure out that there are 9 dimes and 18 nickels.

  29. More Word Problems Solve: A wallet contains the same number of pennies, nickels, and dimes. The coins total $1.44. How many of each type of coin does the wallet contain? Since there is the same number of each type of coin, I can use one variable to stand for each: number of pennies:  p number of nickels:  p number of dimes:  p The value of the coins is the number of cents for each coin times the number of that type of coin, so: value of pennies: 1p value of nickels:  5p value of dimes:  10p The total value is $1.44, so I'll add the above, set equal to 144 cents, and 1p + 5p + 10p = 144 16p = 144 p = 9 There are nine of each type of coin in the wallet.

  30. More Word Problems Solve: In three more years, Miguel's grandfather will be six times as old as Miguel was last year. When Miguel's present age is added to his grandfather's present age, the total is 68. How old is each one now? This exercise refers not only to their present ages, but also to both their ages last year and their ages in three years, so labeling will be very important. I will label Miguel's present age as "m" and his grandfather's present age as "g". Then m + g = 68. Miguel's age "last year" was m – 1. His grandfather's age "in three more years" will be g + 3. The grandfather's "age three years from now" is six times Miguel's "age last year" or, in math: g + 3 = 6(m – 1) This gives me two equations with two variables: m + g = 68 g + 3 = 6(m – 1) Solving the first equation, I get m = 68 – g. (Note: It's okay to solve for "g = 68 – m", too. The problem will work out a bit differently in the middle, but the answer will be the same at the end.) I'll plug "68 – g" into the second equation in place of "m": g + 3 = 6m – 6 g + 3 = 6(68 – g) – 6 g + 3 = 408 – 6g – 6 g + 3 = 402 – 6gg + 6g = 402 – 3 7g = 399 g = 57 Since "g" stands for the grandfather's current age, then the grandfather is 57 years old. Since m + g = 68, then m = 11, and Miguel is presently eleven years old.

  31. Line of Best Fit Example: y = a + bx Used for: Linear regression attempts to model the relationship between two variables by fitting a linear equation to observed data x is the independent or predictor variable and y is the dependent or response variable.  To find a and b we follow the steps: • The sum of the x--  Sx • The sum of the y--  Sy • The sum of the squares of   x--  Sx2 • The sum of the products of x and y--  Sxy 

  32. Works Cited • Math TV.com. Facebook, n.d. Web. 15 May 2010. <http://www.mathtv.com/>. • Quick Math. Web Mathmatica, n.d. Web. 15 May 2010. <http://www.quickmath.com/>.

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