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Exam Review

Learn how to solve equations with one variable by following these steps. Includes examples and properties of equality.

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Exam Review

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  1. Exam Review Created by Noah Freund 14 May 2010 Algebra 1

  2. Solving 1st Power Equations with One Variable

  3. Solving 1st Power Equations in One Variable The object of 1st power equations is to find the value of the variable (x,a,f). Here are the steps to get there: - Find a way to get the variable by its self on one side of the equal sign by doing the opposite of what is placed on in the equation (we will do more of this in the commutative property) x + 3=7  x + 3 - 3=7 - 3  x=4

  4. 1st Power Equations (cont.) Sometimes you may need to do more than just add or subtract, you will sometimes need to divide and multiply. Just keep doing the opposite of what the equation says. x * 3 = 27 y/2 = 12  x * 3 / 3 = 27 / 3  x = 9  y/2 * 2 = 12 * 2  y = 24

  5. 1st Power Equations (cont. more) Sometimes, you may have an equation with the variable on both sides of the equation. Make sure you get both variables on one side. x * 2 = 2 - x  (x + x) * 2 = 2 - x + x  2x * 2 = 2  2x = 1  x = ½

  6. 1st Power Equations (cont.) Sometimes, in special occasions, you will get a answer where the variables cancel out each other. If the both sides of the equal sign are equal, then any real number can be the variable. If not, then it’s nulset. x – 3 = x – 4 + 1 x – 3 = x – 4   -3 = -4 + 1  -3 = -3 {allreals}  -3 -4 

  7. 1st Power Equations (cont. some more) Other times, there will be fractions in the equation containing the variable. Just multiply by the denominator in order for the variable to be a whole number, therefore, you can factor out the value of the variable.

  8. 1st Power Equations (cont. even more) Sometimes, the exponent will be in the denominator of a fraction. When this happens, you multiply the equation by the variable in the denominator.

  9. Properties

  10. Addition Property (of Equality) If X = Y, then X+5 = Y+5 Multiplication Property (of Equality) If X = Y, then X*3 = Y*3

  11. Reflexive Property (of Equality) Simply, X = X Symmetric Property (of Equality) If X = Y, then Y = X Transitive Property (of Equality) If X = Y and Y = Z, then X = Z

  12. Associative Property of Addition If (X+Y)+Z = B, then (Z+X)+Y = B Associative Property of Multiplication If (X*Y)*Z = A, then (Z*X)*Y = A

  13. Commutative Property of Addition (X+Y+Z) = A, so (Z+X+Y) = A Commutative Property of Multiplication (X*Y*Z) = A, so (Z*X*Y) = A

  14. Distributive Property (of Multiplication over Addition) If X*(Y+Z) = A, then XY+XZ = A

  15. Prop of Opposites or Inverse Property of Addition X + -X = 0 Prop of Reciprocals or Inverse Prop. of Multiplication

  16. Identity Property of Addition X+0 = X Identity Property of Multiplication X*1 = X

  17. Multiplicative Property of Zero X*0 = 0 Closure Property of Addition If X and Y is a real numbers then X + Y = Z, then Z is a real number Closure Property of Multiplication If X and Y is a real numbers then X + Y = Z, then Z is a real number

  18. Product of Powers Property XY * AB = (A*X)Y+B Power of a Product Property (X*Y)Z = XZ *YZ Power of a Power Property (X2)2 = (X)2*2 = X4

  19. Quotient of Powers Property Power of a Quotient Property

  20. Zero Power Property If (X+Y)*(A-B) = 0 Then (X+Y) = 0 or (A – B) = 0 So, X =-Y and/or A = B

  21. Negative Power Property

  22. Zero Product Property X0 = 1

  23. Product of Roots Property Quotient of Roots Property

  24. Root of a Power Property Power of a Root Property

  25. Quiz Time! Give the name of the property shown. Click when you’re ready to see the answer. 1. A = B, B = C, so A = C Answer: Transitive Property of Equality

  26. More Quiz! Look at the problem and give the name of the property shown. Click when you’re ready to see the answer. 2. X * 1 = X Answer: Identity Property of Multiplication

  27. Solving 1st Power Inequalities with One Variable.

  28. Solving Inequalities with one Variable Solving Inequalities is just like solving equaions. The difference here is that there is either a ‘greater’ or ‘less than’ sign instead of an equal sign. Equation: x-3 = 2 Inequality: x-3 > 2

  29. -2 If you multiply or divide by a negative, you must reverse the inequality sign. -2x < 4 x > -2 Solution Set: {x: x > -2} Graph of the Solution:

  30. 6 -4 • Open endpoint for these symbols: > < • Closed endpoint for these symbols: ≥ or≤ • Conjunction must satisfy both conditions • Conjunction = “AND” {x: -4 < x ≤ 6} Click to see solution graph

  31. 3 -2 • Open endpoint for these symbols: > < • Closed endpoint for these symbols: ≥ or≤ • Disjunction must satisfy either one or both of the conditions • Disjunction = “OR” {x: x < -2 or x ≥ 3} Click to see solution graph

  32. 3 -2 • Watch for special cases • No solutions that work: Answer is Ø • Every number works: Answer is {reals} • Disjunction in same direction: answer is one arrow {x: x > -2 or x ≥ 3} Click to see solution graph

  33. Ø • Watch for special cases • No solutions that work: Answer is Ø • Every number works: Answer is {reals} • Disjunction in same direction: answer is one arrow {x: -x < -2 and -5x ≥ 15} Click to see solution

  34. 3 -4 Quiz Time! YOU try this problem x - 7 > -10 or -2x < 8 Click to see solution and graph x > 3 or x < -4

  35. Linear Equations with Two Variables

  36. Standard and Point-Slope Form The equation; 4x - 2y = -4 is in standard form.You can tell because of the way it is ordered. Standard form will always be; Ax + By = C The equation y = 2x + 4 is the same equation, only in Point-Slope form. The point slope form always goes; y = mx + b. This way is profitable, because it tells you the slope and the y-intercept. m = slope b = y-intercept

  37. Slopes If it is a positive slope, the line will rise up from left to right and if negative it will run down Slopes are in a form where the rise (vertical) is on top (do this first) and the run (horizontal) on the bottom. What would you move if the slope was 1/5? You would move 1 up and 5 to the right

  38. How to Find Intercepts Intercepts are the places where the line crosses the y-axis or x-axis. To find these in an equation, you set the variable you are not looking for to zero. EX: 3x – 2y = 25 3x – 2y = 25  3(0) – 2y = 25  -2y = 25  y = -12.5  3x – 2(0) = 25  3x = 25  x = 8.33333....

  39. How To Graph When graphing, it’s easier to use the point slope form, since the slope and y-intercept are given to you already. 2x – 3y = 3  -3y = -2x + 3  y = 2/3x + 0 Hint: If there is no value of number for this place the only intercepts the origin.

  40. How To Graph (cont.) Now that it is in point slope form, you can graph it. y = 2/3x + 0 Remember! Rise over Run! Start at the origin... And move up 2 spaces and right 3 spaces until you have a line! Right here Right 3 Up 2

  41. Linear Systems

  42. Substitution Method When given an equation with 2 different variables, you can use this method. Knowing the value of one variable, you can substitute that one for its value. x + 3 = y – 7 and 2x + y = 40 x + 3 = y – 7 is the same as x = y – 10, so if we plug it into the other equation, it will look like; 2(y – 10) + y = 40 2y + y – 20 = 40  3y = 60 y = 20 That being said; x = 20 – 10  x = 10

  43. Addition or Subtraction Method This method takes the two equations and adds or subtracts them together until one of the variables has cancelled out. 3x + 6y = 22 and 6x + 8y = 46 Multiply this one by two so we can subtract the ‘x’ variable out. 2(3x + 6y = 22)  6x + 12y = 22

  44. Addition and Subtraction (cont.) Then subtract the two equations together, 6x + 12y = 22 -6x + 8y = 46 0x + 4y = 68  y = 17 We then plug it into an equation to get the other variable 3x + 6(17) = 22  3x + 102 = 22  3x = -80  x = 26 and 2/3

  45. Consistency is when two equations are exactly the same. The Substitution method would work best. EX: 3x + y = 5 and 3x + y = 5  they are both the same Dependent is when the equations are in relation with each other. They are multiples of each other or sums or differences of each other. If an equation is Independent, then you should use the addition subtraction method EX: 3x + y = 5 and 6x + 2y = 10  are dependent because the second one is the first multiplied by two.

  46. Factoring

  47. Factoring Factoring is using methods to make problems more simple. Using various properties, you can cut down on the amount of ink you use while making it easier to multiply by other things.

  48. Factoring: Reverse Foil Just like foiling, but the opposite. You take a trinomial and turn it into a set of binomials x2 + 3x + 4  (x + 4)(x – 1) Because… x*x = x2 x*-1 = -x x*4 = 4x 4*-1 = -4 Firsts Outsides Insides Lasts

  49. Factoring: PST PST is short for perfect square trinomial. You take a PST and change it into 2 binomials. (x2 – 36)  (x – 6)(x + 6) Because... x*x = x2 x*6 = 6x x*-6 = -6x 6*-6 = -36

  50. Factoring: GCF GCF is short for Greatest Common Factor. You take the greatest number (or glob) you can from a equation. (2x + 6)*(3z + 7)  2(x + 3)* z(3 + 7)

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