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
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?
Material
Chair
Sofa
Wood
4 boxes
3 boxes
Stuffing
Fabric
1 box
2 boxes

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

51. Equations in 3 variables

52. Equations in 3 variables

53. Equations in 3 variables

54. p.173 #22-33

55. Solving Systems of Linear Equations in Three Variables—3.6

56. Solutions 1 solution Infinite Solutions No Solutions
An ordered triple where all 3 planes intersect
Infinite Solutions
All 3 planes intersect to form a line
No Solutions
All 3 planes do not intersect
All 3 planes do not intersect at a common point or line

57. What does this look like graphically?

61. Should we solve graphically
Probably not…
Tough to be accurate
Difficult to find equations and coordinates in 3-D
So….
Solve algebraically

62. Solving Systems

63. Solving Systems

64. p.181 #12, 13, 17-20