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# Math 1300: Section 4-1 Review: Systems of Linear Equations in Two Variables

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

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### Math 1300: Section 4-1 Review: Systems of Linear Equations in Two Variables

1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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