Your SlideShare is downloading. ×
0
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Geometry Section 0-8 11-12
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Text the download link to your phone
Standard text messaging rates apply

Geometry Section 0-8 11-12

480

Published on

System

System

Published in: Education, Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
480
On Slideshare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
2
Comments
0
Likes
0
Embeds 0
No embeds

Report content
Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
No notes for slide

Transcript

  • 1. Section 0-8 Systems of EquationsMonday, September 19, 2011
  • 2. Essential Question How do you use graphing, substitution, and elimination to solve systems of linear equations?Monday, September 19, 2011
  • 3. Vocabulary 1. System of Equations: 2. Substitution: 3. Elimination:Monday, September 19, 2011
  • 4. Vocabulary 1. System of Equations: Two or more equations with the same two variables that you solve at the same time 2. Substitution: 3. Elimination:Monday, September 19, 2011
  • 5. Vocabulary 1. System of Equations: Two or more equations with the same two variables that you solve at the same time 2. Substitution: Plugging in a number or expression for a variable 3. Elimination:Monday, September 19, 2011
  • 6. Vocabulary 1. System of Equations: Two or more equations with the same two variables that you solve at the same time 2. Substitution: Plugging in a number or expression for a variable 3. Elimination: Using addition and multiplication to eliminate parts of a system to achieve a solutionMonday, September 19, 2011
  • 7. Types of SolutionsMonday, September 19, 2011
  • 8. Types of Solutions One solution: a pointMonday, September 19, 2011
  • 9. Types of Solutions One solution: a point No solutions: parallel linesMonday, September 19, 2011
  • 10. Types of Solutions One solution: a point No solutions: parallel lines Infinitely many solutions on the line: same lineMonday, September 19, 2011
  • 11. Example 1 Solve the system by graphing. y xMonday, September 19, 2011
  • 12. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 13. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 14. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 15. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 16. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 17. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 18. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 19. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 20. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 21. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 22. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 23. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 24. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 25. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 26. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 27. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 28. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 29. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 30. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 31. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 32. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 33. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 34. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 35. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 36. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 37. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 38. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 39. Example 1 Solve the system by graphing. y (3, 2) ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 40. Example 1 Solve the system by graphing. y (3, 2) ⎧ y = 2x − 4 Check: ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  • 41. Example 1 Solve the system by graphing. y (3, 2) ⎧ y = 2x − 4 Check: ⎪ ⎨ x ⎪ y = −x +5 ⎩ 2 = 2(3)− 4Monday, September 19, 2011
  • 42. Example 1 Solve the system by graphing. y (3, 2) ⎧ y = 2x − 4 Check: ⎪ ⎨ x ⎪ y = −x +5 ⎩ 2 = 2(3)− 4 2 = 6− 4Monday, September 19, 2011
  • 43. Example 1 Solve the system by graphing. y (3, 2) ⎧ y = 2x − 4 Check: ⎪ ⎨ x ⎪ y = −x +5 ⎩ 2 = 2(3)− 4 2 = 6− 4 2 = −3+5Monday, September 19, 2011
  • 44. Example 1 Solve the system by graphing. y (3, 2) ⎧ y = 2x − 4 Check: ⎪ ⎨ x ⎪ y = −x +5 ⎩ 2 = 2(3)− 4 2 = 6− 4 2 = −3+5Monday, September 19, 2011
  • 45. Solve a System of Equations by SubstitutionMonday, September 19, 2011
  • 46. Solve a System of Equations by Substitution 1. Solve one equation for one variable (your choice)Monday, September 19, 2011
  • 47. Solve a System of Equations by Substitution 1. Solve one equation for one variable (your choice) 2. Substitute the expression from the equation into the other equationMonday, September 19, 2011
  • 48. Solve a System of Equations by Substitution 1. Solve one equation for one variable (your choice) 2. Substitute the expression from the equation into the other equation 3. Solve for the variable and substitute back into the original equation to find the other variableMonday, September 19, 2011
  • 49. Solve a System of Equations by Substitution 1. Solve one equation for one variable (your choice) 2. Substitute the expression from the equation into the other equation 3. Solve for the variable and substitute back into the original equation to find the other variable 4. Rewrite your answer as an ordered pair and check it!Monday, September 19, 2011
  • 50. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩Monday, September 19, 2011
  • 51. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9Monday, September 19, 2011
  • 52. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 10x +(−x + 9) =12xMonday, September 19, 2011
  • 53. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 10x +(−x + 9) =12x 9x + 9 =12xMonday, September 19, 2011
  • 54. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 10x +(−x + 9) =12x 9x + 9 =12x 9 = 3xMonday, September 19, 2011
  • 55. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 10x +(−x + 9) =12x 9x + 9 =12x 9 = 3x x =3Monday, September 19, 2011
  • 56. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 3+ y = 9 10x +(−x + 9) =12x 9x + 9 =12x 9 = 3x x =3Monday, September 19, 2011
  • 57. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 3+ y = 9 10x +(−x + 9) =12x y =6 9x + 9 =12x 9 = 3x x =3Monday, September 19, 2011
  • 58. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 3+ y = 9 10x +(−x + 9) =12x y =6 9x + 9 =12x Check: 9 = 3x x =3Monday, September 19, 2011
  • 59. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 3+ y = 9 10x +(−x + 9) =12x y =6 9x + 9 =12x Check: 9 = 3x 3+ 6 = 9 x =3Monday, September 19, 2011
  • 60. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 3+ y = 9 10x +(−x + 9) =12x y =6 9x + 9 =12x Check: 9 = 3x 3+ 6 = 9 x =3 10(3)+ 6 =12(3)Monday, September 19, 2011
  • 61. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 3+ y = 9 10x +(−x + 9) =12x y =6 9x + 9 =12x Check: 9 = 3x 3+ 6 = 9 x =3 10(3)+ 6 =12(3) 30 + 6 = 36Monday, September 19, 2011
  • 62. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 3+ y = 9 10x +(−x + 9) =12x y =6 9x + 9 =12x Check: (3, 6) 9 = 3x 3+ 6 = 9 x =3 10(3)+ 6 =12(3) 30 + 6 = 36Monday, September 19, 2011
  • 63. Solve by EliminationMonday, September 19, 2011
  • 64. Solve by Elimination 1. Choose a variable to eliminate (your choice).Monday, September 19, 2011
  • 65. Solve by Elimination 1. Choose a variable to eliminate (your choice). 2. Make the coefficients of that variable opposite. You might need to multiply to do this. Then combine equations.Monday, September 19, 2011
  • 66. Solve by Elimination 1. Choose a variable to eliminate (your choice). 2. Make the coefficients of that variable opposite. You might need to multiply to do this. Then combine equations. 3. Solve for the remaining variable.Monday, September 19, 2011
  • 67. Solve by Elimination 1. Choose a variable to eliminate (your choice). 2. Make the coefficients of that variable opposite. You might need to multiply to do this. Then combine equations. 3. Solve for the remaining variable. 4. Plug back into an original equation to find the other variable.Monday, September 19, 2011
  • 68. Solve by Elimination 1. Choose a variable to eliminate (your choice). 2. Make the coefficients of that variable opposite. You might need to multiply to do this. Then combine equations. 3. Solve for the remaining variable. 4. Plug back into an original equation to find the other variable. 5. Check and rewrite the answer.Monday, September 19, 2011
  • 69. Example 3 Solve by combining the equations ⎧7x + 2 y = 5 ⎪ ⎨ ⎪ 2x + 3 y = 16 ⎩Monday, September 19, 2011
  • 70. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16 ⎩Monday, September 19, 2011
  • 71. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩(Monday, September 19, 2011
  • 72. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15Monday, September 19, 2011
  • 73. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 −4x − 6 y = −32Monday, September 19, 2011
  • 74. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 −4x − 6 y = −32Monday, September 19, 2011
  • 75. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 −4x − 6 y = −32 17x = −17Monday, September 19, 2011
  • 76. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 −4x − 6 y = −32 17x = −17 17 17Monday, September 19, 2011
  • 77. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 −4x − 6 y = −32 17x = −17 17 17 x = −1Monday, September 19, 2011
  • 78. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 17x = −17 17 17 x = −1Monday, September 19, 2011
  • 79. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 17 17 x = −1Monday, September 19, 2011
  • 80. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 +2 +2 17 17 x = −1Monday, September 19, 2011
  • 81. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 +2 +2 17 17 3 y = 18 x = −1Monday, September 19, 2011
  • 82. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 +2 +2 17 17 3 y = 18 x = −1 3 3Monday, September 19, 2011
  • 83. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 +2 +2 17 17 3 y = 18 x = −1 3 3 y=6Monday, September 19, 2011
  • 84. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 +2 +2 17 17 3 y = 18 x = −1 3 3 y=6Monday, September 19, 2011
  • 85. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 7(−1) + 2(6) = 5 −2 + 3 y = 16 17x = −17 +2 +2 17 17 3 y = 18 x = −1 3 3 y=6Monday, September 19, 2011
  • 86. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 7(−1) + 2(6) = 5 −2 + 3 y = 16 +2 −7 + 12 = 5 17x = −17 +2 17 17 3 y = 18 x = −1 3 3 y=6Monday, September 19, 2011
  • 87. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 7(−1) + 2(6) = 5 −2 + 3 y = 16 +2 −7 + 12 = 5 17x = −17 +2 17 17 3 y = 18 2(−1) + 3(6) = 16 x = −1 3 3 y=6Monday, September 19, 2011
  • 88. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 7(−1) + 2(6) = 5 −2 + 3 y = 16 +2 −7 + 12 = 5 17x = −17 +2 17 17 3 y = 18 2(−1) + 3(6) = 16 x = −1 3 3 −2 + 18 = 16 y=6Monday, September 19, 2011
  • 89. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 7(−1) + 2(6) = 5 −2 + 3 y = 16 +2 −7 + 12 = 5 17x = −17 +2 17 17 3 y = 18 2(−1) + 3(6) = 16 x = −1 3 3 −2 + 18 = 16 y=6 (−1, 6)Monday, September 19, 2011
  • 90. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩Monday, September 19, 2011
  • 91. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7Monday, September 19, 2011
  • 92. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2(2 y +7)+ 4 y = −14Monday, September 19, 2011
  • 93. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2(2 y +7)+ 4 y = −14 −4 y −14 + 4 y = −14Monday, September 19, 2011
  • 94. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2(2 y +7)+ 4 y = −14 −4 y −14 + 4 y = −14 −14 = −14Monday, September 19, 2011
  • 95. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2(2 y +7)+ 4 y = −14 −4 y −14 + 4 y = −14 −14 = −14 What’s going on here?Monday, September 19, 2011
  • 96. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2 y = −x +7 −2(2 y +7)+ 4 y = −14 −4 y −14 + 4 y = −14 −14 = −14 What’s going on here?Monday, September 19, 2011
  • 97. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2 y = −x +7 −2(2 y +7)+ 4 y = −14 1 7 y= x− −4 y −14 + 4 y = −14 2 2 −14 = −14 What’s going on here?Monday, September 19, 2011
  • 98. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2 y = −x +7 4 y = 2x −14 −2(2 y +7)+ 4 y = −14 1 7 y= x− −4 y −14 + 4 y = −14 2 2 −14 = −14 What’s going on here?Monday, September 19, 2011
  • 99. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2 y = −x +7 4 y = 2x −14 −2(2 y +7)+ 4 y = −14 1 7 1 7 y= x− y= x− −4 y −14 + 4 y = −14 2 2 2 2 −14 = −14 What’s going on here?Monday, September 19, 2011
  • 100. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2 y = −x +7 4 y = 2x −14 −2(2 y +7)+ 4 y = −14 1 7 1 7 y= x− y= x− −4 y −14 + 4 y = −14 2 2 2 2 −14 = −14 These are the same lines! What’s going on here?Monday, September 19, 2011
  • 101. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2 y = −x +7 4 y = 2x −14 −2(2 y +7)+ 4 y = −14 1 7 1 7 y= x− y= x− −4 y −14 + 4 y = −14 2 2 2 2 −14 = −14 These are the same lines! What’s going on here? Infinitely many solutions on the line.Monday, September 19, 2011
  • 102. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2 y = −x +7 4 y = 2x −14 −2(2 y +7)+ 4 y = −14 1 7 1 7 y= x− y= x− −4 y −14 + 4 y = −14 2 2 2 2 −14 = −14 These are the same lines! What’s going on here? Infinitely many solutions on the line.Monday, September 19, 2011
  • 103. Example 4 Solve each system of equations. Check your solution. ⎧2x −7 y = −2 ⎪ b. ⎨ ⎪−4x +14 y = 3 ⎩Monday, September 19, 2011
  • 104. Example 4 Solve each system of equations. Check your solution. ⎧2x −7 y = −2 ⎪ b. ⎨ ⎪−4x +14 y = 3 ⎩ −7 y = −2x − 2Monday, September 19, 2011
  • 105. Example 4 Solve each system of equations. Check your solution. ⎧2x −7 y = −2 ⎪ b. ⎨ ⎪−4x +14 y = 3 ⎩ −7 y = −2x − 2 2 2 y= x− 7 7Monday, September 19, 2011
  • 106. Example 4 Solve each system of equations. Check your solution. ⎧2x −7 y = −2 ⎪ b. ⎨ ⎪−4x +14 y = 3 ⎩ −7 y = −2x − 2 2 2 y= x− 7 7 ⎛2 2⎞ −4x +14 ⎜ x − ⎟ = 3 ⎝7 7⎠Monday, September 19, 2011
  • 107. Example 4 Solve each system of equations. Check your solution. ⎧2x −7 y = −2 ⎪ b. ⎨ ⎪−4x +14 y = 3 ⎩ −7 y = −2x − 2 2 2 y= x− 7 7 ⎛2 2⎞ −4x +14 ⎜ x − ⎟ = 3 ⎝7 7⎠ −4x + 4x − 4 = 3Monday, September 19, 2011
  • 108. Example 4 Solve each system of equations. Check your solution. ⎧2x −7 y = −2 ⎪ b. ⎨ ⎪−4x +14 y = 3 ⎩ −7 y = −2x − 2 2 2 y= x− 7 7 ⎛2 2⎞ −4x +14 ⎜ x − ⎟ = 3 ⎝7 7⎠ −4x + 4x − 4 = 3 −4 = 3Monday, September 19, 2011
  • 109. Example 4 Solve each system of equations. Check your solution. ⎧2x −7 y = −2 ⎪ b. ⎨ ⎪−4x +14 y = 3 ⎩ −7 y = −2x − 2 2 2 14 y = 4x +3 y= x− 7 7 ⎛2 2⎞ −4x +14 ⎜ x − ⎟ = 3 ⎝7 7⎠ −4x + 4x − 4 = 3 −4 = 3Monday, September 19, 2011
  • 110. Example 4 Solve each system of equations. Check your solution. ⎧2x −7 y = −2 ⎪ b. ⎨ ⎪−4x +14 y = 3 ⎩ −7 y = −2x − 2 2 2 14 y = 4x +3 y= x− 7 7 2 3 ⎛2 2⎞ y= x+ −4x +14 ⎜ x − ⎟ = 3 7 14 ⎝7 7⎠ −4x + 4x − 4 = 3 −4 = 3Monday, September 19, 2011
  • 111. Example 4 Solve each system of equations. Check your solution. ⎧2x −7 y = −2 ⎪ b. ⎨ ⎪−4x +14 y = 3 ⎩ −7 y = −2x − 2 2 2 14 y = 4x +3 y= x− 7 7 2 3 ⎛2 2⎞ y= x+ −4x +14 ⎜ x − ⎟ = 3 7 14 ⎝7 7⎠ These lines are parallel. −4x + 4x − 4 = 3 −4 = 3Monday, September 19, 2011
  • 112. Example 4 Solve each system of equations. Check your solution. ⎧2x −7 y = −2 ⎪ b. ⎨ ⎪−4x +14 y = 3 ⎩ −7 y = −2x − 2 2 2 14 y = 4x +3 y= x− 7 7 2 3 ⎛2 2⎞ y= x+ −4x +14 ⎜ x − ⎟ = 3 7 14 ⎝7 7⎠ These lines are parallel. −4x + 4x − 4 = 3 There are no solutions. −4 = 3Monday, September 19, 2011
  • 113. When solving a system you get:Monday, September 19, 2011
  • 114. When solving a system you get: One solution when:Monday, September 19, 2011
  • 115. When solving a system you get: One solution when: No solutions when:Monday, September 19, 2011
  • 116. When solving a system you get: One solution when: No solutions when: An infinite number of solutions on the line when:Monday, September 19, 2011
  • 117. Problem SetMonday, September 19, 2011
  • 118. Problem Set p. P18 #1-15 all “I have failed many times, and that’s why I am a success.” - Michael JordanMonday, September 19, 2011

×