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Systems in Two Variables
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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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Transcript of "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
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