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3.2 – Solving Systems of Eqs. Algebraically

3.2 – Solving Systems of Eqs. Algebraically. 3.2 – Solving Systems of Eqs. Algebraically. Recall that when solving graphically, solution is point of intersection. 3.2 – Solving Systems of Eqs. Algebraically. Recall that when solving graphically, solution is point of intersection.

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3.2 – Solving Systems of Eqs. Algebraically

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  1. 3.2 – Solving Systems of Eqs. Algebraically

  2. 3.2 – Solving Systems of Eqs. Algebraically • Recall that when solving graphically, solution is point of intersection.

  3. 3.2 – Solving Systems of Eqs. Algebraically • Recall that when solving graphically, solution is point of intersection. Substitution Method

  4. 3.2 – Solving Systems of Eqs. Algebraically • Recall that when solving graphically, solution is point of intersection. Substitution Method Ex. 1 Use substitution to solve the system of equations. x + 2y = 8 ½x – y = 18

  5. 3.2 – Solving Systems of Eqs. Algebraically • Recall that when solving graphically, solution is point of intersection. Substitution Method Ex. 1 Use substitution to solve the system of equations. x + 2y = 8 ½x – y = 18 • Solve 1st eq. for variable

  6. 3.2 – Solving Systems of Eqs. Algebraically • Recall that when solving graphically, solution is point of intersection. Substitution Method Ex. 1 Use substitution to solve the system of equations. x + 2y = 8 ½x – y = 18 • Solve 1st eq. for variable (whichever is easiest)

  7. 3.2 – Solving Systems of Eqs. Algebraically • Recall that when solving graphically, solution is point of intersection. Substitution Method Ex. 1 Use substitution to solve the system of equations. x + 2y = 8 ½x – y = 18 • Solve 1st eq. for variable (whichever is easiest) x + 2y = 8

  8. 3.2 – Solving Systems of Eqs. Algebraically • Recall that when solving graphically, solution is point of intersection. Substitution Method Ex. 1 Use substitution to solve the system of equations. x + 2y = 8 ½x – y = 18 • Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y

  9. 3.2 – Solving Systems of Eqs. Algebraically • Recall that when solving graphically, solution is point of intersection. Substitution Method Ex. 1 Use substitution to solve the system of equations. x + 2y = 8 ½x – y = 18 • Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y x = -2y + 8

  10. 3.2 – Solving Systems of Eqs. Algebraically • Recall that when solving graphically, solution is point of intersection. Substitution Method Ex. 1 Use substitution to solve the system of equations. x + 2y = 8 ½x – y = 18 • Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y x = -2y + 8 • Substitute in and solve for other variable!

  11. 3.2 – Solving Systems of Eqs. Algebraically • Recall that when solving graphically, solution is point of intersection. Substitution Method Ex. 1 Use substitution to solve the system of equations. x + 2y = 8 ½x – y = 18 • Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y x = -2y + 8 • Substitute in and solve for other variable! ½x – y = 18

  12. 3.2 – Solving Systems of Eqs. Algebraically • Recall that when solving graphically, solution is point of intersection. Substitution Method Ex. 1 Use substitution to solve the system of equations. x + 2y = 8 ½x – y = 18 • Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y x = -2y + 8 • Substitute in and solve for other variable! ½x – y = 18

  13. 3.2 – Solving Systems of Eqs. Algebraically • Recall that when solving graphically, solution is point of intersection. Substitution Method Ex. 1 Use substitution to solve the system of equations. x + 2y = 8 ½x – y = 18 • Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y x = -2y + 8 • Substitute in and solve for other variable! ½x – y = 18

  14. 3.2 – Solving Systems of Eqs. Algebraically • Recall that when solving graphically, solution is point of intersection. Substitution Method Ex. 1 Use substitution to solve the system of equations. x + 2y = 8 ½x – y = 18 • Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y x = -2y + 8 • Substitute in and solve for other variable! ½x – y = 18 ½(-2y + 8) – y = 18

  15. 3.2 – Solving Systems of Eqs. Algebraically • Recall that when solving graphically, solution is point of intersection. Substitution Method Ex. 1 Use substitution to solve the system of equations. x + 2y = 8 ½x – y = 18 • Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y x = -2y + 8 • Substitute in and solve for other variable! ½x – y = 18 ½(-2y + 8) – y = 18

  16. 3.2 – Solving Systems of Eqs. Algebraically • Recall that when solving graphically, solution is point of intersection. Substitution Method Ex. 1 Use substitution to solve the system of equations. x + 2y = 8 ½x – y = 18 • Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y x = -2y + 8 • Substitute in and solve for other variable! ½x – y = 18 ½(-2y + 8) – y = 18 -y + 4 – y = 18

  17. 3.2 – Solving Systems of Eqs. Algebraically • Recall that when solving graphically, solution is point of intersection. Substitution Method Ex. 1 Use substitution to solve the system of equations. x + 2y = 8 ½x – y = 18 • Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y x = -2y + 8 • Substitute in and solve for other variable! ½x – y = 18 ½(-2y + 8) – y = 18 -y + 4 – y = 18 -2y + 4 = 18

  18. 3.2 – Solving Systems of Eqs. Algebraically • Recall that when solving graphically, solution is point of intersection. Substitution Method Ex. 1 Use substitution to solve the system of equations. x + 2y = 8 ½x – y = 18 • Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y x = -2y + 8 • Substitute in and solve for other variable! ½x – y = 18 ½(-2y + 8) – y = 18 -y + 4 – y = 18 -2y + 4 = 18 -2y = 14

  19. 3.2 – Solving Systems of Eqs. Algebraically • Recall that when solving graphically, solution is point of intersection. Substitution Method Ex. 1 Use substitution to solve the system of equations. x + 2y = 8 ½x – y = 18 • Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y x = -2y + 8 • Substitute in and solve for other variable! ½x – y = 18 ½(-2y + 8) – y = 18 -y + 4 – y = 18 -2y + 4 = 18 -2y = 14 y = -7

  20. Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y x = -2y + 8 • Substitute in and solve for other variable! ½x – y = 18 ½(-2y + 8) – y = 18 -y + 4 – y = 18 -2y + 4 = 18 -2y = 14 y = -7 • Substitute into equation from 1) and solve for x.

  21. Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y x = -2y + 8 • Substitute in and solve for other variable! ½x – y = 18 ½(-2y + 8) – y = 18 -y + 4 – y = 18 -2y + 4 = 18 -2y = 14 y = -7 • Substitute into equation from 1) and solve for x. x = -2y + 8

  22. Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y x = -2y + 8 • Substitute in and solve for other variable! ½x – y = 18 ½(-2y + 8) – y = 18 -y + 4 – y = 18 -2y + 4 = 18 -2y = 14 y = -7 • Substitute into equation from 1) and solve for x. x = -2y + 8 x = -2(-7) + 8

  23. Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y x = -2y + 8 • Substitute in and solve for other variable! ½x – y = 18 ½(-2y + 8) – y = 18 -y + 4 – y = 18 -2y + 4 = 18 -2y = 14 y = -7 • Substitute into equation from 1) and solve for x. x = -2y + 8 x = -2(-7) + 8 x = 14 + 8

  24. Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y x = -2y + 8 • Substitute in and solve for other variable! ½x – y = 18 ½(-2y + 8) – y = 18 -y + 4 – y = 18 -2y + 4 = 18 -2y = 14 y = -7 • Substitute into equation from 1) and solve for x. x = -2y + 8 x = -2(-7) + 8 x = 14 + 8 x = 22

  25. Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y x = -2y + 8 • Substitute in and solve for other variable! ½x – y = 18 ½(-2y + 8) – y = 18 -y + 4 – y = 18 -2y + 4 = 18 -2y = 14 y = -7 • Substitute into equation from 1) and solve for x. x = -2y + 8 x = -2(-7) + 8 x = 14 + 8 x = 22

  26. Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y x = -2y + 8 • Substitute in and solve for other variable! ½x – y = 18 ½(-2y + 8) – y = 18 -y + 4 – y = 18 -2y + 4 = 18 -2y = 14 y = -7 • Substitute into equation from 1) and solve for x. x = -2y + 8 x = -2(-7) + 8 x = 14 + 8 x = 22 (22

  27. Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y x = -2y + 8 • Substitute in and solve for other variable! ½x – y = 18 ½(-2y + 8) – y = 18 -y + 4 – y = 18 -2y + 4 = 18 -2y = 14 y = -7 • Substitute into equation from 1) and solve for x. x = -2y + 8 x = -2(-7) + 8 x = 14 + 8 x = 22 (22

  28. Solve 1st eq. for variable (whichever is easiest) x + 2y = 8 - 2y - 2y x = -2y + 8 • Substitute in and solve for other variable! ½x – y = 18 ½(-2y + 8) – y = 18 -y + 4 – y = 18 -2y + 4 = 18 -2y = 14 y = -7 • Substitute into equation from 1) and solve for x. x = -2y + 8 x = -2(-7) + 8 x = 14 + 8 x = 22 (22,-7)

  29. Elimination Method

  30. Elimination Method Ex. 2 Use the elimination method to solve the system of equations.

  31. Elimination Method Ex. 2 Use the elimination method to solve the system of equations. a. 4a + 2b = 15 2a + 2b = 7

  32. Elimination Method Ex. 2 Use the elimination method to solve the system of equations. a. 4a + 2b = 15 2a + 2b = 7 • Make numbers of 1 of the variables the same number with opposite signs, then add the equations together

  33. Elimination Method Ex. 2 Use the elimination method to solve the system of equations. a. 4a + 2b = 15 2a + 2b = 7 • Make numbers of 1 of the variables the same number with opposite signs, then add the equations together 4a + 2b = 15

  34. Elimination Method Ex. 2 Use the elimination method to solve the system of equations. a. 4a + 2b = 15 2a + 2b = 7 • Make numbers of 1 of the variables the same number with opposite signs, then add the equations together 4a + 2b = 15 (-1)[2a + 2b = 7]

  35. Elimination Method Ex. 2 Use the elimination method to solve the system of equations. a. 4a + 2b = 15 2a + 2b = 7 • Make numbers of 1 of the variables the same number with opposite signs, then add the equations together 4a + 2b = 15 -2a - 2b = -7

  36. Elimination Method Ex. 2 Use the elimination method to solve the system of equations. a. 4a + 2b = 15 2a + 2b = 7 • Make numbers of 1 of the variables the same number with opposite signs, then add the equations together 4a+ 2b = 15 -2a- 2b = -7

  37. Elimination Method Ex. 2 Use the elimination method to solve the system of equations. a. 4a + 2b = 15 2a + 2b = 7 • Make numbers of 1 of the variables the same number with opposite signs, then add the equations together 4a+ 2b = 15 -2a- 2b = -7 2a + 0 = 8

  38. Elimination Method Ex. 2 Use the elimination method to solve the system of equations. a. 4a + 2b = 15 2a + 2b = 7 • Make numbers of 1 of the variables the same number with opposite signs, then add the equations together 4a + 2b = 15 -2a - 2b = -7 2a = 8

  39. Elimination Method Ex. 2 Use the elimination method to solve the system of equations. a. 4a + 2b = 15 2a + 2b = 7 • Make numbers of 1 of the variables the same number with opposite signs, then add the equations together 4a + 2b = 15 -2a - 2b = -7 2a = 8 a = 4

  40. Elimination Method Ex. 2 Use the elimination method to solve the system of equations. a. 4a + 2b = 15 2a + 2b = 7 • Make numbers of 1 of the variables the same number with opposite signs, then add the equations together 4a + 2b = 15 -2a - 2b = -7 2a = 8 a = 4 • Plug 4 into first eq. and solve for b.

  41. Elimination Method Ex. 2 Use the elimination method to solve the system of equations. a. 4a + 2b = 15 2a + 2b = 7 • Make numbers of 1 of the variables the same number with opposite signs, then add the equations together 4a + 2b = 15 -2a - 2b = -7 2a = 8 a = 4 • Plug 4 into first eq. and solve for b. 4(4) + 2b = 15

  42. Elimination Method Ex. 2 Use the elimination method to solve the system of equations. a. 4a + 2b = 15 2a + 2b = 7 • Make numbers of 1 of the variables the same number with opposite signs, then add the equations together 4a + 2b = 15 -2a - 2b = -7 2a = 8 a = 4 • Plug 4 into first eq. and solve for b. 4(4) + 2b = 15 16 + 2b = 15

  43. Elimination Method Ex. 2 Use the elimination method to solve the system of equations. a. 4a + 2b = 15 2a + 2b = 7 • Make numbers of 1 of the variables the same number with opposite signs, then add the equations together 4a + 2b = 15 -2a - 2b = -7 2a = 8 a = 4 • Plug 4 into first eq. and solve for b. 4(4) + 2b = 15 16 + 2b = 15 2b = -1

  44. Elimination Method Ex. 2 Use the elimination method to solve the system of equations. a. 4a + 2b = 15 2a + 2b = 7 • Make numbers of 1 of the variables the same number with opposite signs, then add the equations together 4a + 2b = 15 -2a - 2b = -7 2a = 8 a = 4 • Plug 4 into first eq. and solve for b. 4(4) + 2b = 15 16 + 2b = 15 2b = -1 b = -½

  45. Elimination Method Ex. 2 Use the elimination method to solve the system of equations. a. 4a + 2b = 15 2a + 2b = 7 • Make numbers of 1 of the variables the same number with opposite signs, then add the equations together 4a + 2b = 15 -2a - 2b = -7 2a = 8 a = 4 • Plug 4 into first eq. and solve for b. 4(4) + 2b = 15 16 + 2b = 15 2b = -1 b = -½, So the lines intersect at (4, -½)

  46. b. 3x – 7y = -14 5x + 2y = 45

  47. b. 3x – 7y = -14 5x + 2y = 45 • Make numbers of 1 of the variables the same number with opposite signs, then add the equations together

  48. b. 3x – 7y = -14 5x + 2y = 45 • Make numbers of 1 of the variables the same number with opposite signs, then add the equations together 3x – 7y = -14 5x + 2y = 45

  49. b. 3x – 7y = -14 5x + 2y = 45 • Make numbers of 1 of the variables the same number with opposite signs, then add the equations together (2)[3x – 7y = -14] (7)[5x + 2y = 45]

  50. b. 3x – 7y = -14 5x + 2y = 45 • Make numbers of 1 of the variables the same number with opposite signs, then add the equations together 6x – 14y = -28 35x + 14y = 315

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