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Advanced Algebra Chapter 3

Advanced Algebra Chapter 3

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Advanced Algebra Chapter 3

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  1. Advanced Algebra Chapter 3 Systems Of Linear Equations and Inequalities

  2. Solving Systems by Graphing—3.1

  3. System of two linear equations: • Solution: The ordered pair (x, y) that satisfies both equations • Where the equations intersect

  4. Check to see if the following point is a solution of the linear system: (2, 2)

  5. Check to see if the following point is a solution of the linear system: (0, -1)

  6. Solving Graphically

  7. Solving Graphically

  8. Solving Graphically

  9. Solving Graphically

  10. Interpretation • The graphs intersect at 1 specific point • Exactly one solution • The graph is a single line • Infinitely many solutions • The graphs never intersect • No solutions

  11. p.142 #11-49 Odd

  12. Solving Systems Algebraically—3.2

  13. Substitution Method • 1.) Solve one of the equations for one of the variables • 2.) Substitute the expression into the other equation • 3.) Find the value of the variable • 4.) Use this value in either of the original equations to find the 2nd variable

  14. Substitution Method

  15. Substitution Method

  16. Substitution Method

  17. Substitution Method

  18. p.152#11-19

  19. Solving by Linear Combination • 1.) Multiply 1 or both equations by a constant to get similar coefficients • 2.) Add or subtract the revised equations to get 1 equation with only 1 variable • Something must cancel! • 3.) Solve for the variable • 4.) Use this value to solve for the 2nd variable • 5.) Smile 

  20. Linear Combinations

  21. Linear Combinations

  22. Linear Combinations

  23. Linear Combinations

  24. p153 #23-31

  25. Pop Quiz!!Graphing Linear Inequalities • Graph the following:

  26. Graphing Linear Inequalities

  27. Graphing Linear Inequalities

  28. Graphing Linear Inequalities

  29. Solving Systems of Linear Inequalities—3.3

  30. Systems • Solution of two linear equations: • Ordered pair • Solution of two linear inequalities • Infinite Solutions • An entire region

  31. Solving Linear Inequalities

  32. Solving Linear Inequalities

  33. Solving Linear Inequalities

  34. Solving Linear Inequalities

  35. p.159 #13-49 EOO

  36. Optimization—3.4

  37. Optimization • Optimization • Finding the maximum or minimum value of some quantity • Linear Programming: • Optimizing linear functions • Objective Function: • What we are trying to maximize or minimize • The linear inequalities making up the program: constraints • Points contained in the graph: feasible region

  38. Optimal Solution • The optimal Solution (minimum or maximum value) must occur at a vertex of the feasible region • If the region is bounded, a minimum and maximum value must occur within the feasible region

  39. Solving: Finding min and max • Objective Function: • Constraints:

  40. Solving: Finding min and max • Objective Function: • Constraints:

  41. Solving: Finding min and max • Objective Function: • Constraints:

  42. A Furniture Manufacturer makes chairs and sofas from prepackaged parts. The table below gives the number of packages of wood parts, stuffing, and material required for each chair and sofa. The packages are delivered weekly and manufacturer has room to store 1300 packages of wood parts, 2000 packages of stuffing, and 800 packages of fabric. The manufacturer profits $200 per chair and $350 per sofa. How many of each should they make per week?

  43. Writing Inequalities • Optimization: • Constraints:

  44. p.166 #9-15, 21

  45. Graphing in Three Dimensions—3.5

  46. z-axis • Ordered triple • Octants

  47. (-2, 1, 6)

  48. (3, 4, 0)

  49. (0, 4, -2)

  50. Linear Equations ax + by + cz = d An ordered triple is a solution of the equation The graph of an equation of three variables is the graph of all it’s solutions -The graph will be a plane