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Algebra Exam Review Project

Algebra Exam Review Project. Created by Cal Larson. First Power Equations. It is simple find out what X is equivalent to. You can add, subtract, multiply and/or divide REMEMBER WHAT YOU DO ON ONE SIDE OF THE EQUATION YOU DO TO THE OTHER!!!!!!!!!!!!!!!!!!!!!!! 5=x+3 x=2. More questions.

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Algebra Exam Review Project

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  1. Algebra Exam Review Project Created by Cal Larson

  2. First Power Equations • It is simple find out what X is equivalent to. • You can add, subtract, multiply and/or divide • REMEMBER WHAT YOU DO ON ONE SIDE OF THE EQUATION YOU DO TO THE OTHER!!!!!!!!!!!!!!!!!!!!!!! • 5=x+3 • x=2

  3. More questions • 5x+3=4x • X=-3 • 1/2x+15=20 • x+30=40 • x=10 • 5(x+2)=15 • 5x+10=15 • 5x=5 • X=1

  4. Weird ones • 5x+6x+14=4x+7(x+2) • 11x+14=4x+7x+14 • 11x+14=11x+14 • The answer is x=all real numbers or everything • 2/x=5 • 2=10x • 1/5=x

  5. Properties

  6. Addition Property (of Equality) Example: If a = b, then a + c = b + c Multiplication Property (of Equality) Example: If a = b, then ca = cb

  7. Example: If “a” is a real number, then a = a Reflexive Property (of Equality) Symmetric Property (of Equality) Example: If a = b, then b = a. Transitive Property (of Equality) Example: If a = b, and b = c, then a = c.

  8. Example: (a + b) + c = a + (b + c) Associative Property of Addition Associative Property of Multiplication Example: (ab)c = a(bc)

  9. Commutative Property of Addition Example: a + b = b + a Commutative Property of Multiplication Example: ab = ba

  10. Distributive Property (of Multiplication over Addition) Example: a(b + c) = ab + ac

  11. Prop of Opposites or Inverse Property of Addition Example: -(a + b) = (-a) + (-b) Prop of Reciprocals or Inverse Prop. of Multiplication Example: a • 1/a = 1 and 1/a • a = 1

  12. Identity Property of Addition Example: If a + 0 = a, then 0 + a = a. Identity Property of Multiplication Example: If a • 1 = a, then 1• a = a.

  13. Example: If a • 0 = 0, then 0 • a = 0. Multiplicative Property of Zero Closure Property of Addition Example: a + b is a unique real number Closure Property of Multiplication Example: abisa unique real number

  14. Example:am • an = am+n Product of Powers Property Power of a Product Property Example: (ab)m = ambm Power of a Power Property Example: (am)n = amn

  15. Quotient of Powers Property Power of a Quotient Property Example: ( )m =

  16. Example: If any number to the 0 power is 1 x0=1 Zero Power Property Negative Power Property Example: If an exponent is to a negative number then the number is the denominator over 1 X-5= 1/x5

  17. Zero Product Property Example: If ab = 0, then a = 0 or b = 0.

  18. Example: Product of Roots Property Quotient of Roots Property The square root of a divided by the square root of b equals the square root of a over b

  19. Root of a Power Property Example: r2=s2 r=s r=-s

  20. First power inequalities • This means means x is greater than or equal to 5 • This means x is less then or equal to 11 • This means x is greater than to 15 • This means x is less than -5 • They are mostly the same however they will not be equal

  21. First power inequalities cont • IF YOU MULTIPLY OR DIVIDE BY A NEGATIVE NUMBER THEN SWITCH THE SIGN!!!!!!!! • I.E. Divide by –x and switch the inequality sign

  22. Graphing • To graph you have to make a line graph and make is so x is equal or greater than five. • There should be a dark dot for greater than or equal however my math program won’t let me do it • To graph you make the line graph so x will be smaller than -5

  23. Graphing cont • It is the same with just greater than or less than but there is no black dot just a circle on the graph • If there are two equations and you use the word and then you shade in the overlapping area or the line • If there are 2 equations and they have the word or then you just graph the two on the same line.

  24. Quiz

  25. Quiz cont • This one might be a little weird • And • Null set

  26. Another question • Have fun with this one The answer is all real numbers

  27. Linear equations with 2 variables • This is not a fun unit I hated it and I’m sure you will hate it also, have fun 

  28. Simple stuff • Y=mx+b is very simple • Y is the outcome m is the slope x is the input and b is the y-intercept • Y=3x-5 is an example of Y=mx+b • Y is the output 3 is the slope and -5 is the y intercept • A 3 slope means the point slides over 1 and up 3 • The y intercept is where the line touches the y axis

  29. More simple stuff • The Y intercept will always start be 0,b • Y=mx+b is standard form • To find the slope for a straight line you need to take the difference of the rise (X) over the difference over the run (Y). • For example if the coordinates are 3,4 and 6,8 • 4-8/3-6 • -4/-3 • 4/3

  30. Even more simple stuff • The slope is 4/3 • Point slope form is when you have the slope and you have a point on the graph • Y-y1=m(x-x1) • If the slope is 2 and the point on the graph is 0,3 • Y-0=2(x-3) • Y=2x-6 • Now it is in standard form • Some problems will ask for it in standard form while others will ask for it in point slope form

  31. Some stuff requiring a little thinking • How do you find the y and x intercepts? • 6x+2y=12 • To find the y intercept you set Y to 0 and solve to find the x intercept set x to 0 and solve • 6x+0y=12 • X=2 the y intercept is 0,2 • 0x+2y=12 • y=6 • The x intercept is 6,0

  32. Why do I care about the x and y intercepts? • They give you the slope and y intercept!! • This allows you to find the equation of a line in standard form • Example from the last problem • 6/2=3 the slope is 3 • Y=3x+2

  33. Quiz Time!! • What is the slope and y intercept of the equation Y=5x-3? • Slope is 5 and y intercept is 0,-3 • Put this equation in standard form • The coordinates are -3,1 and -2,3 • 1-3/-3+2 • -2/-1 • 2 • The slope is 2

  34. More fun quizzes • Y+3=2(x-2) • Y+3=2x-4 • Y=2x-1 • Find the x and y intercepts for the equation 5x+2y=20 • 5(0)+2y=20 • Y=10 • x intercept is 10

  35. The last page of quizzes • 5x+y(0)=10 • X=2 • y intercept is 0,2

  36. Linear systems • In this unit of slideshows I will show you how to solve equations with y and x as variables • The first method is the substitution method • This method works when in one part of the equation has the coefficient of x or y = 1 • 2y+x=15 • 2y+3x=20 • X=-2y+15 • 2y+3(-2y+15)=20

  37. Linear systems cont. • 2y-6y-45=20 • -4y=-25 • Y=25/4 • Now enter y into the original equation • 50/4+x=15 • X=1 1/2 • Next is the elimination method • You try to eliminate one variable by multiplying so one variable is the opposite of the other variable

  38. Examples of the elmination • X+2y=10 • X+y=7 • Multiply by -1 • -x-y=-7 • Then “add” the two equations • Y=3 • X+6=10 • X=4

  39. Quiz Time! • X=y+2 • 2x+2y=10 • 2(y+2)+2y=10 • 4y+4=10 • 4y=10 • Y=2.5 • 2.5+2=x • x=5.5

  40. Second question • 2x+3y=15 • 3x+3y=12 • -2x-3y=-15 • 3x+3y=12 • X=-3 • -6+3y=15 • Y=7

  41. Factoring • I will cover this briefly because it was our last unit • The sum/difference of cubes is (a+b)3 • (a+b)(a2+ab+b2) • The grouping 3 by 1 is (a+b)2+c2 • ((a+b)+c)((a+b)+c) • A perfect square trinomial is (x+b)2 • X2+b2+b2

  42. More factoring • Dots or difference of two squares • (x-5)(x+2) • x2-3-10 • The GCF is greatest common factor • 15x2+15x+30 • 15(x2+x+2) • Grouping 2 by 2 is • x2+2x+x3+2x2 • X(x+2)+x2(x+2) • (x+x2)(x+2)

  43. Rational expressions • A rational number is a number expressed as quotient of two integers • The denominator has to have a variable in it

  44. How to do them • It is a lot easier than it seems • For addition just add the numerator and denominator and just simplify • For X2/x you simplify so the answer is just x • For addition or subtraction of two rational expressions you make the signs one and just continue • x/y+x/y=x+x/y+y • The same applies for subtraction

  45. Multiplication • It is the same thing as addition • (x/y)*(x/y)=(2x/2y) • Division is different first you do the reciprocal of one number then you multiply them • (x/y)*(x/y)=(x/y)/(y/x)

  46. Quadratics • For strait factoring you set the equation to 0 • X2+10x+25=0 • (x+5)(x+5) the • You want to set the answer to zero so you make x be the opposite of the constant • The answer is x=-5 • Another way is taking the root of both sides • 25=x2 • Take the square root of both sides and you get your answer • 5=x

  47. More quadratics • Completing the square • X2-6x-3=0 • X2-6x =3 • Add (b/2)2 to both sides • x2-6x+9=12 • (x-3)2=12 • Get the square root and simplify • X-3=2 Square root of 3

  48. The quadratic formula

  49. More of the quadratic formula • It should b2 but my math program won’t let me do that • X2+7x+10 • -21/2 • The discriminant tells me if the equation will work or not • The discriminant is b2-4ac

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