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Equations with Number Solutions

Learn how to solve equations using the numbers {-2, -1, 0, 1, 2}. Understand equivalent equations and determine the best first step (BFS) to simplify and solve equations faster.

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Equations with Number Solutions

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  1. Bellwork 9/24 Using the numbers {-2, -1, 0, 1, 2} Solve the following equations. • 1.) -2x = 2 • 2.) 3x = x + 4 • 3.) x = 4/x

  2. Using the numbers {-2, -1, 0, 1, 2} Solve the following equations. • 1.) -2x = 2 2.) 3x = x + 4 • -2(-2) = 2 3(-2) = -2 + 4 • -2(-1) = 2 3(-1) = -1 + 4 • -2(0) = 2 3(0) = 0 + 4 • -2(1) = 2 3(1) = 1 + 4 • -2(2) = 2 3(2) = 2 + 4 F F T F F F F F F T

  3. Using the numbers {-2, -1, 0, 1, 2} Solve the following equations. • 3.) x = 4/x • -2 = 4/-2 • -1 = 4/-1 • 0 = 4/0 • 1 = 4/1 • 2 = 4/2 True False False False True

  4. Bellwork 9/24 Using the numbers {-2, -1, 0, 1, 2} Solve the following equations. • -2(-1) = 2 • 2 = 2 Since -1 makes • 1.) -2x = 2 the equation true, it is the solution.

  5. Bellwork 9/24 Using the numbers {-2, -1, 0, 1, 2} Solve the following equations. • 3(2) = 2 + 4 • 6 = 6 Since 2 makes • 2.) 3x = x+4 the equation true, it is the solution.

  6. Bellwork 9/24 Using the numbers {-2, -1, 0, 1, 2} Solve the following equations. • -2 = 4/-2 • -2 = -2 • 3.) x = 4/x AND

  7. Bellwork 9/24 Using the numbers {-2, -1, 0, 1, 2} Solve the following equations. • 2 = 4/2 • 2 = 2 Since 2 & -2 make • 3.) x = 4/x the equation true, they are solutions.

  8. Today’s Objective • To be able to recognize equations that are equivalent to other equations…. • Understanding how to make the BEST FIRST STEP (the BFS)

  9. Take good notes Today This icon, Means to take notes

  10. Suppose ….. (Pizza Pizza) • You and 3 of your friends ordered a pizza. The cost of the pizza was $16.80. How could you write an equation to determine the cost for each of the 4 people?

  11. Techniques and strategies for solving equations. • Up to this point you have been using: a.)Mental Math to solve equations. b.)Substitution to solve equations Now we will look at strategies

  12. Equ Equivalent Equations al • Equivalent Equations are equations that look different, but have the same solutions… • Example: • 2x = 8 and 2x + 1 = 9 Take Notes

  13. We will show these equations to be equivalent with the number (4) they must be equivlalent • 2(4) = 8 2(4) + 1 = 9 • 8 = 8 8 + 1 = 9 • 9 = 9 1.) 2x = 8 2x + 1 = 9 Since both equations are True for 4, T T

  14. Expressions Equations • Equations are made up of two (2) Expressions separated by an equal (=) sign. • Example: 2x + 3x = 30 Equal sign

  15. Equations must maintain a BALANCE What you do to one side must be done to the other

  16. Start with x - 7 = 3 • Add 5 to both sides: x-7+5 = 3+5 • Simplify: x - 2 = 8 • This NEW equation is equivalent to the FIRST equation. Because x = 10

  17. Now Consider x - 7 = 3 • Add 7 to both sides: x-7+7 = 3+7 • Simplify: x = 10 • Of These two examples which one simplifies the equation the most???

  18. Start with a + 3 = 1 • Subtract 3 from : a + 3 - 3 = 1 -3 • Simplify: a = -2 • Why did we subtract 3 from both sides of the equation?? • Subtracting 3 is the BFS

  19. Determining the BEST FIRST STEP (BFS) • The BFS is the step that will simplify the equation in the least amount of steps. • In 1 step equations, it is usually the inverse operation.

  20. Start with 1/2y = 6 • Multiply both sides by 2 : • (2)1/2y = (2)6 • Simplify: y = 12 • Why did we multiply both sides by 2 ?? • Multiplying by 2 is the BFS

  21. Start with -3m = 12 • Divide both sides by -3 : • -3m/-3 = 12/-3 • Simplify: m = -4 • Why did we divide both sides by -3 ?? • Dividing by -3 is the BFS

  22. What does all this mean ? • An Equivalent Equation is obtained when you add, subtract, multiply, or divide the same number to BOTH SIDES of the equation. • BUT REMEMBER,

  23. You can do what ever you want to an equation, as long as you do it to both sides! • So the TRICK is to determine the BFS to make solving equations easier, faster, and simpler

  24. Be Ready to Answer the Following: • 1.) What’s the difference between an Expression and an Equation? • 2.) What separates the 2 expressions of an equation? • 3.) You can do anything you want to an equation as long as _____? • 4.) What’s BFS?

  25. What’s the Best First Step Divide by 3 • 1.) 3x = 6 • 2.) x - 8 = 7 • 3.) a + 5 = 9 • 4.) x/3 = 9 Add 8 Subtract 5 Multiply by 3

  26. What’s the Best First Step Divide by -2 • 5.) -2x = 8 • 6.) 2x = 7 • 7.) ½a = 9 • 8.) -3x = 9 Divide by 2 Mult by 2 Divide by -3

  27. What’s the Best First Step CLT • 9.) 3x - 5x = 6 • 10.) 2x - 8 = 6 • 11.) a + 5 = 9 • 12.) x/3 + 6 = 9 Add 8 Sub 5 Sub 6

  28. Now for the steps • 1.) Simplify each side of the equation. • 2.) Get the Variables together first. • 3.) Then, get the Whole Numbers together. • 4.) Finally, get the variable by itself.

  29. Now you try theseShow all work on page 3 1.) 3x = 6 2.) x - 8 = 7 3.) a + 5 = 9 4.) x/3 = 9

  30. Now you try these • 1.) 3x = 6 • 3x/3 = 6/3 • x = 2 • 2.) x - 8 = 7 • x - 8 + 8 = 7 + 8 • x = 15

  31. Now you try these • 3.) a + 5 = 9 • a + 5 - 5 = 9 - 5 • a = 4 • 4.) x/3 = 9 • (3)x/3 = 9(3) • x = 27

  32. Now back to the Pizza Pizza Problem • You and 3 of your friends ordered a pizza. The cost of the pizza was $16.80. How could you write an equation to determine the cost for each of the 4 people?

  33. Pizza Pizza • Let x = your cost • Then 4x = the cost of all 4 people • So…. 4x = 16.80 • 4x/4 = 16.80/4 Divide both sides by 4 x = 4.20 Classwork

  34. Follow the directions in order • 4x - 5 = 15 • 1.) Add 5 to both sides • 2.) Divide both sides by 4 • 1.) 4x -5 + 5 = 15 + 5 • 4x = 20 • 2.) 4x/4 = 20/4 • x = 5

  35. Follow the directions in order • 1/2x -12 = 18 • 1.) Add 12 to both sides • 2.) Multiply both sides by 2 • 1.) 1/2x - 12 + 12 = 18 + 12 • 1/2x = 30 • 2.) (2)1/2x = 30(2) • x = 60

  36. Classwork • Do worksheet 3.1 • Homework • Page 126 (1-12 & 14-48 ev)

  37. Bellwork (Show all work) 9/27 1.) 4x = -24 2.) x - 9 = 14 3.) a + 15 = 9 4.) x/2 = 12

  38. Now you try these • 1.) 4x = -24 • 4x/4 = -24/4 • x = -6 • 2.) x - 9 = 14 • x - 9 + 9 = 14 + 9 • x = 23

  39. Now you try these • 3.) a + 15 = 9 • a + 15 - 15 = 9 - 15 • a = -6 • 4.) x/2 = 12 • (2)x/2 = 12(2) • x = 24

  40. Bellwork (Show all work) 9/29 1.) -3x = -24 2.) x - 7 = 4 3.) a - 8 = 8 4.) 1/2x = 12

  41. Bellwork (Show all work) • 1.) -3x = -24 • -3x/-3 = -24/-3 • x = 8 • 2.) x - 7 = 4 • x - 7 + 7 = 4 + 7 • x = 11

  42. Bellwork (Show all work) • 3.) a - 8 = 8 • a -8 + 8 = 8 + 8 • a = 16 • 4.) 1/2x = 12 • (2)1/2x = 12(2) • x = 24

  43. Solve the following EquationsShow your work • 4.) -3 = 7 + a • -10 = a • 5.) 10x = 100 • x = 10 • 6.) 256 = 16c • 16 = c • 1.) 4 + t = 13 • t = 9 • 2.) w - 5 =11 • w = 16 • 3.) |-3| + n = 0 • n = -3

  44. Solve the following EquationsShow your work • 10.) x = 2 4 • x = 8 • 11.) c = 15 7 • c = 105 • 7.) 1/2x = -40 • x = -80 • 8.) 3/4L = 75 • L = 100 • 9.) -6x = -36 • x = 6

  45. Follow the directions in order • 4x - 5 = 15 • 1.) Add 5 to both sides • 2.) Divide both sides by 4 • 1.) 4x -5 + 5 = 15 + 5 • 4x = 20 • 2.) 4x/4 = 20/4 • x = 5

  46. Follow the directions in order • 1/2x -12 = 18 • 1.) Add 12 to both sides • 2.) Multiply both sides by 2 • 1.) 1/2x - 12 + 12 = 18 + 12 • 1/2x = 30 • 2.) (2)1/2x = 30(2) • x = 60

  47. homework • Classwork Do worksheet 3.1 ( ) • Homework • page 128 (59-78)

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