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Lecture over Section 4-1 of Barnett's "Finite Mathematics."

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- 1. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Math 1300 Finite Mathematics Section 4.1 Review: Systems of Linear Equations in Two Variables Jason Aubrey Department of Mathematics University of Missouri university-logo Jason Aubrey Math 1300 Finite Mathematics
- 2. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Deﬁnition (Systems of Equations in Two Variables) Given the linear system ax + by = h cx + dy = k A pair of numbers x = x0 , y = y0 [also written as an ordered pair (x0 , y0 ) is said to be a solution of the system if each equation is satisﬁed by the pair. The set of all such ordered pairs is called the solution set for the system. To solve a system is to ﬁnd its solution set. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 3. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Deﬁnition (Systems of Equations in Two Variables) Given the linear system ax + by = h cx + dy = k A pair of numbers x = x0 , y = y0 [also written as an ordered pair (x0 , y0 ) is said to be a solution of the system if each equation is satisﬁed by the pair. The set of all such ordered pairs is called the solution set for the system. To solve a system is to ﬁnd its solution set. We will consider three methods for solving such systems: graphing, substitution, and elimination by addition. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 4. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve the linear system by graphing: 3x − y = 2 x + 2y = 10 10 8 6 4 2 −4 −2 2 4 6 8 10 university-logo −2 Jason Aubrey Math 1300 Finite Mathematics
- 5. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve the linear system by graphing: 3x − y = 2 x + 2y = 10 10 8 6 4 2 −4 −2 2 4 6 8 10 university-logo −2 Jason Aubrey Math 1300 Finite Mathematics
- 6. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve the linear system by graphing: 3x − y = 2 x + 2y = 10 10 8 6 4 2 −4 −2 2 4 6 8 10 university-logo −2 Jason Aubrey Math 1300 Finite Mathematics
- 7. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve the linear system by graphing: 3x − y = 2 x + 2y = 10 10 8 6 (2, 4) 4 2 −4 −2 2 4 6 8 10 university-logo −2 Jason Aubrey Math 1300 Finite Mathematics
- 8. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Deﬁnition (Systems of Linear Equations: Basic Terms) A system of linear equations is consistent if it has one or more solutions and inconsistent if no solutions exist. Furthermore, a consistent system is said to be independent if it has exactly one solution and dependent if it has more than one solution. Two systems of equations are equivalent if they have the same solution set. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 9. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Theorem (Possible Solutions to a Linear System) The linear system ax + by = h cx + dy = k university-logo Jason Aubrey Math 1300 Finite Mathematics
- 10. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Theorem (Possible Solutions to a Linear System) The linear system ax + by = h cx + dy = k Must have Exactly one solution (consistent and independent), or university-logo Jason Aubrey Math 1300 Finite Mathematics
- 11. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Theorem (Possible Solutions to a Linear System) The linear system ax + by = h cx + dy = k Must have Exactly one solution (consistent and independent), or No solution (inconsistent), or university-logo Jason Aubrey Math 1300 Finite Mathematics
- 12. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Theorem (Possible Solutions to a Linear System) The linear system ax + by = h cx + dy = k Must have Exactly one solution (consistent and independent), or No solution (inconsistent), or Inﬁnitely many solutions (consistent and dependent). university-logo Jason Aubrey Math 1300 Finite Mathematics
- 13. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Theorem (Possible Solutions to a Linear System) The linear system ax + by = h cx + dy = k Must have Exactly one solution (consistent and independent), or No solution (inconsistent), or Inﬁnitely many solutions (consistent and dependent). There are no other possibilities. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 14. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Graph the equations and ﬁnd the coordinates of any points where two or more lines intersect. Discuss the nature of the solution set. x − 2y = −6 2x + y = 8 x + 2y = −2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 15. Systems in Two Variables Graphing Substitution Elimination by Addition Applications 6 4 2 −4 −2 0 2 4 −2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 16. Systems in Two Variables Graphing Substitution Elimination by Addition Applications 6 x − 2y = −6 4 2 −4 −2 0 2 4 −2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 17. Systems in Two Variables Graphing Substitution Elimination by Addition Applications 6 x − 2y = −6 4 2x + y = 8 2 −4 −2 0 2 4 −2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 18. Systems in Two Variables Graphing Substitution Elimination by Addition Applications 6 x − 2y = −6 4 2x + y = 8 x + 2y = −2 2 −4 −2 0 2 4 −2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 19. Systems in Two Variables Graphing Substitution Elimination by Addition Applications 6 (2, 4) x − 2y = −6 4 2x + y = 8 x + 2y = −2 2 −4 −2 0 2 4 −2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 20. Systems in Two Variables Graphing Substitution Elimination by Addition Applications 6 (2, 4) x − 2y = −6 4 2x + y = 8 (−2, 2) x + 2y = −2 2 −4 −2 0 2 4 −2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 21. Systems in Two Variables Graphing Substitution Elimination by Addition Applications 6 (2, 4) x − 2y = −6 4 2x + y = 8 (−2, 2) x + 2y = −2 2 No point lies on all three lines. So, no so- −4 −2 0 2 4 ( 14 , − 4 ) 3 3 lution to this system. −2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 22. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve the linear system by substitution: 2x + y = 6 x − y = −3 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 23. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve the linear system by substitution: 2x + y = 6 x − y = −3 2x + y = 6 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 24. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve the linear system by substitution: 2x + y = 6 x − y = −3 2x + y = 6 y = 6 − 2x now substitute: university-logo Jason Aubrey Math 1300 Finite Mathematics
- 25. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve the linear system by substitution: 2x + y = 6 x − y = −3 2x + y = 6 y = 6 − 2x now substitute: x − y = −3 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 26. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve the linear system by substitution: 2x + y = 6 x − y = −3 2x + y = 6 y = 6 − 2x now substitute: x − y = −3 x − (6 − 2x) = −3 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 27. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve the linear system by substitution: 2x + y = 6 x − y = −3 2x + y = 6 y = 6 − 2x now substitute: x − y = −3 x − (6 − 2x) = −3 3x − 6 = −3 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 28. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve the linear system by substitution: 2x + y = 6 x − y = −3 2x + y = 6 y = 6 − 2x now substitute: x − y = −3 x − (6 − 2x) = −3 3x − 6 = −3 x =1 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 29. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve the linear system by substitution: 2x + y = 6 x − y = −3 2x + y = 6 y = 6 − 2x now substitute: x − y = −3 x − (6 − 2x) = −3 3x − 6 = −3 x =1 Substituting we obtain y = 4. So, the solution is the point (1, 4). university-logo Jason Aubrey Math 1300 Finite Mathematics
- 30. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Graphing and substitution work well for systems involving two variables. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 31. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Graphing and substitution work well for systems involving two variables. However, neither is easily extended to larger systems. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 32. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Graphing and substitution work well for systems involving two variables. However, neither is easily extended to larger systems. Elimination by addition is the most important method of solution. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 33. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Graphing and substitution work well for systems involving two variables. However, neither is easily extended to larger systems. Elimination by addition is the most important method of solution. It readily generalizes to larger systems and forms the basis for computer-based solution methods. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 34. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Theorem (Operations that Produce Equivalent Systems) A system of linear equations is transformed into an equivalent system if university-logo Jason Aubrey Math 1300 Finite Mathematics
- 35. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Theorem (Operations that Produce Equivalent Systems) A system of linear equations is transformed into an equivalent system if (A) Two equations are interchanged. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 36. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Theorem (Operations that Produce Equivalent Systems) A system of linear equations is transformed into an equivalent system if (A) Two equations are interchanged. (B) An equation is multiplied by a nonzero constant. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 37. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Theorem (Operations that Produce Equivalent Systems) A system of linear equations is transformed into an equivalent system if (A) Two equations are interchanged. (B) An equation is multiplied by a nonzero constant. (C) A constant multiple of one equation is added to another equation. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 38. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve the linear system using elimination by addition 3u − 2v = 12 7u + 2v = 8 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 39. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve the linear system using elimination by addition 3u − 2v = 12 7u + 2v = 8 3u − 2v = 12 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 40. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve the linear system using elimination by addition 3u − 2v = 12 7u + 2v = 8 3u − 2v = 12 +7u + 2v = 8 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 41. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve the linear system using elimination by addition 3u − 2v = 12 7u + 2v = 8 3u − 2v = 12 +7u + 2v = 8 10u + 0 = 20 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 42. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve the linear system using elimination by addition 3u − 2v = 12 7u + 2v = 8 3u − 2v = 12 +7u + 2v = 8 10u + 0 = 20 u=2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 43. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve the linear system using elimination by addition 3u − 2v = 12 7u + 2v = 8 3u − 2v = 12 +7u + 2v = 8 10u + 0 = 20 u=2 Substituting, we obtain v = −3. So the solution is (2,-3) university-logo Jason Aubrey Math 1300 Finite Mathematics
- 44. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve using elimination by addition. 3x − 2y = 8 2x + 5y = −1 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 45. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve using elimination by addition. 3x − 2y = 8 2x + 5y = −1 We multiply the ﬁrst equation by 5 and the bottom equation by 2 and then add. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 46. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve using elimination by addition. 3x − 2y = 8 2x + 5y = −1 We multiply the ﬁrst equation by 5 and the bottom equation by 2 and then add. 15x − 10y = 40 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 47. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve using elimination by addition. 3x − 2y = 8 2x + 5y = −1 We multiply the ﬁrst equation by 5 and the bottom equation by 2 and then add. 15x − 10y = 40 4x + 10y = −2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 48. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve using elimination by addition. 3x − 2y = 8 2x + 5y = −1 We multiply the ﬁrst equation by 5 and the bottom equation by 2 and then add. 15x − 10y = 40 4x + 10y = −2 19x = 38 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 49. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve using elimination by addition. 3x − 2y = 8 2x + 5y = −1 We multiply the ﬁrst equation by 5 and the bottom equation by 2 and then add. 15x − 10y = 40 4x + 10y = −2 19x = 38 1 Now we multiply both sides of this last equation by 19 to obtain x = 2. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 50. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: Solve using elimination by addition. 3x − 2y = 8 2x + 5y = −1 We multiply the ﬁrst equation by 5 and the bottom equation by 2 and then add. 15x − 10y = 40 4x + 10y = −2 19x = 38 1 Now we multiply both sides of this last equation by 19 to obtain x = 2. Then we substitute back into either of the two original equations to obtain y = −1. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 51. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Example: At $4.80 per bushel, the annual supply for soybeans in the Midwest is 1.9 billion bushels and the annual demand is 2.0 billion bushels. When the price increases to $5.10 per bushel, the annual supply increases to 2.1 billion bushels and the annual demand decreases to 1.8 billion bushels. Assume that the supply and demand equations are linear. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 52. Systems in Two Variables Graphing Substitution Elimination by Addition Applications (a) Find the supply equation. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 53. Systems in Two Variables Graphing Substitution Elimination by Addition Applications (a) Find the supply equation. We wish to ﬁnd an equation of the form p = mq + b where p represents unit price and q represents quantity demanded. We have two points (q, p) on the graph of the supply equation: (1.9, 4.80) and (2.1, 5.10). university-logo Jason Aubrey Math 1300 Finite Mathematics
- 54. Systems in Two Variables Graphing Substitution Elimination by Addition Applications (a) Find the supply equation. We wish to ﬁnd an equation of the form p = mq + b where p represents unit price and q represents quantity demanded. We have two points (q, p) on the graph of the supply equation: (1.9, 4.80) and (2.1, 5.10). The slope of this line is: university-logo Jason Aubrey Math 1300 Finite Mathematics
- 55. Systems in Two Variables Graphing Substitution Elimination by Addition Applications (a) Find the supply equation. We wish to ﬁnd an equation of the form p = mq + b where p represents unit price and q represents quantity demanded. We have two points (q, p) on the graph of the supply equation: (1.9, 4.80) and (2.1, 5.10). The slope of this line is: 5.10 − 4.80 0.3 3 m= = = 2.1 − 1.9 0.2 2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 56. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Therefore the equation of this line is university-logo Jason Aubrey Math 1300 Finite Mathematics
- 57. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Therefore the equation of this line is 3 p − 4.8 = (q − 1.9) 2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 58. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Therefore the equation of this line is 3 p − 4.8 = (q − 1.9) 2 3 3 p = q − (1.9) + 4.8 2 2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 59. Systems in Two Variables Graphing Substitution Elimination by Addition Applications Therefore the equation of this line is 3 p − 4.8 = (q − 1.9) 2 3 3 p = q − (1.9) + 4.8 2 2 3 p = q + 1.95 2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 60. Systems in Two Variables Graphing Substitution Elimination by Addition Applications (b) Find the demand equation. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 61. Systems in Two Variables Graphing Substitution Elimination by Addition Applications (b) Find the demand equation. Again, we want an equation of the form p = mq + b. We have two points on the graph of the demand equation: (2.0, 4.80) and (1.8, 5.10). university-logo Jason Aubrey Math 1300 Finite Mathematics
- 62. Systems in Two Variables Graphing Substitution Elimination by Addition Applications (b) Find the demand equation. Again, we want an equation of the form p = mq + b. We have two points on the graph of the demand equation: (2.0, 4.80) and (1.8, 5.10). The slope of this line is: 5.10 − 4.80 0.3 3 m= =− =− 1.8 − 2.0 0.2 2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 63. Systems in Two Variables Graphing Substitution Elimination by Addition Applications (b) Find the demand equation. Again, we want an equation of the form p = mq + b. We have two points on the graph of the demand equation: (2.0, 4.80) and (1.8, 5.10). The slope of this line is: 5.10 − 4.80 0.3 3 m= =− =− 1.8 − 2.0 0.2 2 Therefore the equation of this line is 3 p − 4.80 = − (q − 2.0) 2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 64. Systems in Two Variables Graphing Substitution Elimination by Addition Applications (b) Find the demand equation. Again, we want an equation of the form p = mq + b. We have two points on the graph of the demand equation: (2.0, 4.80) and (1.8, 5.10). The slope of this line is: 5.10 − 4.80 0.3 3 m= =− =− 1.8 − 2.0 0.2 2 Therefore the equation of this line is 3 p − 4.80 = − (q − 2.0) 2 3 3 p = − q + (2.0) + 4.80 2 2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 65. Systems in Two Variables Graphing Substitution Elimination by Addition Applications (b) Find the demand equation. Again, we want an equation of the form p = mq + b. We have two points on the graph of the demand equation: (2.0, 4.80) and (1.8, 5.10). The slope of this line is: 5.10 − 4.80 0.3 3 m= =− =− 1.8 − 2.0 0.2 2 Therefore the equation of this line is 3 p − 4.80 = − (q − 2.0) 2 3 3 p = − q + (2.0) + 4.80 2 2 3 p = − q + 7.80 university-logo 2 Jason Aubrey Math 1300 Finite Mathematics
- 66. Systems in Two Variables Graphing Substitution Elimination by Addition Applications (c) Find the equilibrium price and quantity. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 67. Systems in Two Variables Graphing Substitution Elimination by Addition Applications (c) Find the equilibrium price and quantity. We set supply equal to demand and solve: 3 3 − q + 7.80 = q + 1.95 2 2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 68. Systems in Two Variables Graphing Substitution Elimination by Addition Applications (c) Find the equilibrium price and quantity. We set supply equal to demand and solve: 3 3 − q + 7.80 = q + 1.95 2 2 3q = 5.85 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 69. Systems in Two Variables Graphing Substitution Elimination by Addition Applications (c) Find the equilibrium price and quantity. We set supply equal to demand and solve: 3 3 − q + 7.80 = q + 1.95 2 2 3q = 5.85 q = 1.95 billion bushels university-logo Jason Aubrey Math 1300 Finite Mathematics
- 70. Systems in Two Variables Graphing Substitution Elimination by Addition Applications (c) Find the equilibrium price and quantity. We set supply equal to demand and solve: 3 3 − q + 7.80 = q + 1.95 2 2 3q = 5.85 q = 1.95 billion bushels Substituting, we ﬁnd p = − 3 (1.95) + 7.80 = $4.88 2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 71. Systems in Two Variables Graphing Substitution Elimination by Addition Applications (d) Graph the two equations in the same coordinate system and identify the equilibrium point, supply curve, and demand curve. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 72. Systems in Two Variables Graphing Substitution Elimination by Addition Applications f (d) Graph the two equations in the same coordinate system and identify the equilibrium point, supply curve, and demand curve. 6 4 2 0 2 4 6 8 10 12 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 73. Systems in Two Variables Graphing Substitution Elimination by Addition Applications f (d) Graph the two equations in the same coordinate system and identify the equilibrium point, supply curve, and demand curve. 6 4 2 0 2 4 6 8 10 12 g university-logo Jason Aubrey Math 1300 Finite Mathematics
- 74. Systems in Two Variables Graphing Substitution Elimination by Addition Applications f (d) Graph the two equations in the same coordinate system and identify the equilibrium point, supply curve, and demand curve. 6 (1.95, 4.88) 4 2 0 2 4 6 8 10 12 g university-logo Jason Aubrey Math 1300 Finite Mathematics
- 75. Systems in Two Variables Graphing Substitution Elimination by Addition Applications f (d) Graph the two equations in the same coordinate system and identify the equilibrium point, supply curve, and demand curve. Supply Curve 6 (1.95, 4.88) 4 2 0 2 4 6 8 10 12 g university-logo Jason Aubrey Math 1300 Finite Mathematics
- 76. Systems in Two Variables Graphing Substitution Elimination by Addition Applications f (d) Graph the two equations in the same coordinate system and identify the equilibrium point, supply curve, and demand curve. Supply Curve 6 (1.95, 4.88) 4 2 Demand Curve 0 2 4 6 8 10 12 g university-logo Jason Aubrey Math 1300 Finite Mathematics

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