1. SYSTEMS OF EQUATIONS

2. Unit 2 Systems of Equations
A system of equations is two or more equations with the same variables. ex. 3x-2y=8 4x+3y=12 There are several ways to solve systems. 1. solve by graphing 2. solve by substitution 3. solve by linear combination 4. solve using matrices

3. Solving by Graphing
When solving a system by graphing you need to graph each line and look for a point of intersection.
The point (x,y) is the solution to the system.
To be a solution, it must satisfy both equations.

4. Classifying Systems
If a system has at least one solution, then it is called consistent
If a system does not have a solution then it is called inconsistent
If the system is consistent and has one solution, then it is called independent
If the system is consistent and has an infinite number of solutions, then it is called dependent.

5. Solving by Graphing
If the lines intersect at one point, then there is one solution
If the lines do not intersect at all, then there is no solution
If the lines intersect at every point, then there is an infinite number of solutions

6. Examples

7. Examples

8. Solving Algebraically
Substitution Method—
Solve one equation for one variable (best to solve for variable with a coefficient of 1 if possible)
Substitute for the variable you just solved for in first step into the other equation
Once you find the value of one variable, substitute into either equation to find value of other variable

9. Example

10. Example

11. Unique Solutions
When solving by substitution method and the variables cancel out and the statement you are left with is false, like 5=8, then there is no solution to the system
When solving and the variables cancel out and the statement you are left with is true, then there is an infinite # of Solutions (not all reals!)

12. Examples

13. Linear Combination also called addition method or elimination method
Add the two equations together so that one variable cancels out (may need to multiply one or both of the equations by some factor that would cause a variable to cancel
Need to have equation in standard form

14. Examples

15. More examples

16. More Examples

17. Real World Examples Assign variables Set up equations Solve
Ex. The perimeter of a rectangle is 36 inches. The length is twice the width. Find the width.

18. Real World Examples
Ex: Thirty people are going to lunch. It costs $5 for kids and $12 for adults. The total bill without tax was $220. How many kids and adults went to lunch?

19. Systems with 3 equations and 3 variables
Steps:
Take 2 equations and eliminate a variable
Take two different equations and eliminate the SAME variable
Take the 2 new 2 variable equations and eliminate one of those variables
Plug the value into either 2 variable eq. for the value of the 2nd variable
Plug both values into original eq. to find value of 3rd variable

20. Example

21. Work Space
(7,4,-5)

22. More Examples

23. Work Space
(4,-3,-1)

24. One more Example—don’t complain!!!

25. Work Space
(2/3,2,-5)

26. Try one more…

27. (-2,3/5,2)

28. Homework

29. Hw answers

30. System of Linear Inequalities
To graph a linear inequality, you graph the related linear equation and shade the area that contains the solution
< or > symbols have a boundary line that is dotted—if it is ≤ or ≥ then the boundary line is solid
Shade the region that includes x and y values that make the inequality true (test a point)

31. System of Linear Inequalities, cont.
2 or more inequalities
Graph each inequality and shade
Darken where the two shaded regions overlap—this will be the solution to the system

32. Example

33. Example

34. Example

35. Linear Programming
A method used to find optimal solutions such as maximum or minimum profits
Steps:
Assign variables
Determine the constraints (inequalities)
Find the feasible region (area of solution)
Determine the vertices of feasible region
Plug those values into the profit equation(also called objective function)

36. Example
Mr. Farmer wants to plant some corn and wheat and he gets the following statistics from the US Bureau of Census
Crop
Yield per acre
Avg Price
Corn
113.5 bu
$3.15/bu
Soybeans
34.9 bu
$6.80 bu
Wheat
35.8 bu
$4.45 bu
Cotton
540 lb
$0.759/lb
Rice, rough
5621 lb
$0.0865/lb

37. Example continued
Mr. Farmer can have no more 120 acres of corn and wheat
At least 20 and no more than 80 acres of corn
At least 30 acres of wheat
How many acres of each crop should Mr. Farmer plant to maximize the revenue from his harvest?

38. Working through the problem..
Assign Variables
X=acres of corn and y=acres of wheat
List the constraints

39. Graph it

40. Example continued List the vertices Determine Profit equation
Which would yield the most?

41. Another Ex.
A snack bar cooks and sells hamburgers and hot dogs during football games. To stay in business, it must sell at least 10 hamburgers but cannot cook more than 40. It must also sell at least 30 hot dogs but cannot cook more than 70. The snack bar cannot cook more than 90 items total. The profit on a hamburger is $0.33 and on a hot dog it is $ How many of each item should it sell to make the maximum profit?
Profit Equation: __________________________
Constraints:
Answer: _________________________

42. Another Example
As a receptionist for a veterinarian, Sue scheduled appointments. She allots 20 minutes for a routine office visit and 40 minutes for surgery. The vet can not do more than 6 surgeries per day. The office has 7 hours available for appointments. If an office visits costs $55 and most surgeries costs $125, find a combination of office visits and surgeries that will maximize the income the veterinarian practice receives per day.

43. What do you know… Assign variables x=number of office visits
y=number of surgeries
Constraints: 7 hours needs to be in terms of minutes

44. Continued… Graph and determine coordinates of the vertices
You should get (0,0) (0,6)(9,6)(21,0)

45. Continued…. Determine the profit equation $55v + $125s = P
Test the points
Highest profit would be when there are 6 surgeries and 9 visits