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Algebra Exam Review Project

Brush up on algebra concepts with equations, properties, graphing tips, and more for your exam preparation. Practice equations and properties to master algebraic principles efficiently. Learn about graphing techniques for linear equations with two variables.

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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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