2. SOLVING EQUATIONSThe single most important skill in algebraGoal: Find the one value of the variable that makes the sentence true.

3. For instance, if 2x + 3 = 17, then 7 is the only value of x that will make this a true sentence. So x = 7.

4. We can solve equations by doing the OPPOSITE of what has been done to the variable in the problem.If a problem says +, you subtract.If a problem has multiplication, you divide.

5. By doing the opposite, we keep the sides of the equation balanced.

6. As long as you do the SAME thing to both sides of an equation, it will remain balanced.

7. 7x – 13 = 50

8. 7x – 13 = 50 +13 +13 7x = 63

9. 7x – 13 = 50 +13 +13 7x = 63 7 7 x = 9

10. -5x + 7 = 82

11. -5x + 7 = 82 - 7 - 7 -5x = 75

12. -5x + 7 = 82 - 7 - 7 -5x = 75 -5 -5 x = -15

13. 4x + 25 = 13

14. 4x + 25 = 13 - 25 -25 4x = -12

15. 4x + 25 = 13 - 25 -25 4x = -12 4 4 x = -3

16. When you solve equations, you also do the opposite of the order of operations.  Add/subtract first  Then divide/multiply

17. Solve these equations: 4a + 11 = 59 -2b + 13 = 5 5c – 72 = 98 -4d + 11 = 47

18. Solve these equations: 4a + 11 = 59  a = 12 -2b + 13 = 5  b = 4 5c – 72 = 98  c = 34 -4d + 11 = 47  d = -9

19. Solve these equations: 5x – 18 = 40 2y + 73 = 54 -3z + 5 = 1

20. Solve these equations: 5x – 18 = 40  x = 58 / 5 or 11.6 2y + 73 = 54  y = -19 / 2 or -9.5 _ -3z + 5 = 1  z = 4 / 3 or 1.3

21. Some equations are even easier. n + 4 = 13 5x = 35

22. Some equations are even easier. n + 4 = 13 Just subtract 4 … x = 9 5x = 35 Just divide by 5 … x = 7

24. Solve 17 = 3x – 7

25. Solve 17 = 3x – 7 +7 +7 24 = 3x

26. Solve 17 = 3x – 7 +7 +7 24 = 3x 3 3 8 = x

27. Solve 19 – 2x = 104

28. Solve 19 – 2x = 104 -19 -19 -2x = 85

29. Solve 19 – 2x = 104 -19 -19 -2x = 85 -2 -2 x = -85 / 2 or -42.5

30. If you know the basic steps, you can quickly do equations with more difficult numbers using a calculator.

31. 12x + 1794 = 2127

33. 963 – 25x = 704

35. What about this?

36. Fractions mean division, so to cancel, we’ll subtract 13 and then multiply by 7.  n = 63

38. Things that can complicate solving equations …

39. ParenthesesUse distributive property first. Like termsCombine them first.

40. 3(2x – 5) = 27

42. 3(2x – 5) = 27 6x – 15 = 27 6x = 42 x = 7

43. -7(2x – 11) = 98

44. -7(2x – 11) = 98 -14x + 77 = 98 -14x = 21 x = -3 / 2 or -1.5

45. 4p + 3 – 2p + 7 + 5p + 2 = 17

46. 4p + 3 – 2p + 7 + 5p + 2 = 17 7p + 12 = 17 7p = 5 p = 5/ 7

47. 5(3x + 5) – 3(2x – 1) = 145

48. 5(3x + 5) – 3(2x – 1) = 145 15x + 25 – 6x + 3 = 145

49. 5(3x + 5) – 3(2x – 1) = 145 15x + 25 – 6x + 3 = 145 9x + 28 = 145

50. 5(3x + 5) – 3(2x – 1) = 145 15x + 25 – 6x + 3 = 145 9x + 28 = 145 9x = 117 x = 13

51. The goal is always to simplify. Make the problem look like the easy ones we know how to solve.

52. Variable on Both SidesFind the smaller number of the variable, and subtract that on both sides.Solve the remaining problem.

53. 7x – 15 = 2x + 70

54. 7x – 15 = 2x + 70 -2x -2x 5x – 15 = 70

55. 7x – 15 = 2x + 70 -2x -2x 5x – 15 = 70 5x = 85 x = 17

56. 5x + 13 = 7x + 40

57. 5x + 13 = 7x + 40 -5x -5x 13 = 2x + 40

58. 5x + 13 = 7x + 40 -5x -5x 13 = 2x + 40 x = -27 / 2 or -13.5

59. 3(2x + 7) = 3x + 4 + x + 9

60. 3(2x + 7) = 3x + 4 + x + 9 6x + 21 = 4x + 13

61. 3(2x + 7) = 3x + 4 + x + 9 6x + 21 = 4x + 13 2x + 21 = 13 2x = -8 x = -4

62. Yesterday you saw this equation: 3x + 5(4x – 2) – 2(x – 4) = 4(2x – 7 + 3x) Solve it.

63. 3x + 5(4x – 2) – 2(x – 4) = 4(2x – 7 + 3x) 3x + 20x – 10 – 2x + 8 = 8x – 28 + 12x

64. 3x + 5(4x – 2) – 2(x – 4) = 4(2x – 7 + 3x) 3x + 20x – 10 – 2x + 8 = 8x – 28 + 12x 21x – 2 = 20x – 28

65. 3x + 5(4x – 2) – 2(x – 4) = 4(2x – 7 + 3x) 3x + 20x – 10 – 2x + 8 = 8x – 28 + 12x 21x – 2 = 20x – 28 -20x -20x x – 2 = -28 +2 +2 x = -26

67. Special equations 2(3x – 7) = 6x + 11 10x – 15 = 5(2x – 3)

68. 2(3x – 7) = 6x + 11 6x – 14 = 6x + 11 ?????

69. 10x – 15 = 5(2x – 3) 10x – 15 = 10x – 15 ?????

70. When variables cancel out…If you have the exact same thing on both sides (like 8 = 8), the answer is ALL REAL NUMBERS or INFINITELY MANY SOLUTIONS. 10x – 15 = 5(2x – 3) 10x – 15 = 10x – 15 -15 = -15

71. An equation with infinitely many solutions can also be called an IDENTITY.

72. If there is something different on the 2 sides (like 5 = 7), there is NO SOLUTION. 2(3x – 7) = 6x + 11 6x – 14 = 6x + 11 -14 = 11