1. 8.2 Inconsistent and Dependent Systems

2. Objective: To solve dependent and inconsistent systems of equations.
Today, we will learn how to solve systems of equations that have infinite solutions or no solution.
The systems of equations we have solved to this point have always had a unique solution. It is reasonable to assume in real life that there may be certain constraints that limit the solutions to a problem but there may be more than one possible solution. It may be your job as a professional to figure out what solution is best.
Today will just deal with systems of equations, tomorrow we will solve applications.

3. Warning!
We will not be graphing linear equation with 3 variables. I will use the following graphs to demonstrate a point.

4. If you graph a system of equations the intersection is the solution.
x = 1 and y = 4 are solutions to this system.
This system has no solutions.
If you graph a system of equations the intersection is the solution.

5. The graph of a linear equation with 3 variables is a plane

6. If the three planes intersect at a point then the system of equations is consistent independent
(1,2,5)

7. If the system is inconsistent then the equations do not have any common solutions

8. If the system is inconsistent then the equations do not have any common solutions

9. Dependent Systems
Three linear equations with 3 variables can have an infinite number of solutions and each graph can still be unique

10. Inconsistent
No Solutions

11. Give 4 possible solutions

12. Solve the following system
Solve the following system. If it is dependent list the solution in terms of z and give 4 possible answers, if it is inconsistent answer no solution.

13. Solve the following system
Solve the following system. If it is dependent list the solution in terms of z and give 4 possible answers, if it is inconsistent answer no solution.

14. Dependent Systems
If there are fewer equations than variables and the system is consistent then there will be an infinite number of solutions

15. Solve the following system
Solve the following system. If it is dependent list the solution in terms of z and give 3 possible answers, if it is inconsistent answer no solution.
rref then use the math enter to change the decimal to a fraction

16. Solve the following dependent consistent system and list 3 possible answers

17. Page 869 #1-8, 16, 18 If the system has no solution state so
Page #1-8, 16, 18 If the system has no solution state so. If the system has infinite solutions list it as an ordered triple in terms of z then give 3 example solutions.

18. A company sells 3 brands of humidifiers: the Jr
A company sells 3 brands of humidifiers: the Jr. which weighs 10 lbs and comes in an 8 ft3 box; the Momma weighs 20 lbs and comes in an 8 ft3 box; the Big Daddy weighs 60 lbs and comes in a 28 ft3 box. Each delivery van has 248 ft3 of space and can hold a maximum of 440 lbs. In order for the van to be fully loaded how many of each model should it carry?
1. Find the question to identify the variables
There are three different types of humidifiers so we need three variables
J= # of Jr’s
M=# of Mommas
D=# of Big Daddy's

19. A company sells 3 brands of humidifiers: the Jr
A company sells 3 brands of humidifiers: the Jr. which weighs 10 lbs and comes in an 8 ft3 box; the Momma weighs 20 lbs and comes in an 8 ft3 box; the Big Daddy weighs 60 lbs and comes in a 28 ft3 box. Each delivery van has 248 ft3 of space and can hold a maximum of 440 lbs. In order for the van to be fully loaded how many of each model should it carry.
J= # of Jrs
M=# of Mommas
D=# of Big Daddies Jr
Momma
Big Daddy
Weight
Volume of box
10 lbs
20 lbs
60 lbs
440 8 8 28
248

20. A company sells 3 brands of humidifiers: the Jr
A company sells 3 brands of humidifiers: the Jr. which weighs 10 lbs and comes in an 8 ft3 box; the Momma weighs 20 lbs and comes in an 8 ft3 box; the Big Daddy weighs 60 lbs and comes in a 28 ft3 box. Each delivery van has 248 ft3 of space and can hold a maximum of 440 lbs. In order for the van to be fully loaded how many of each model should it carry.
J= # of Jrs
M=# of Mommas
D=# of Big Daddies

21. A company sells 3 humidifiers models : the Jr
A company sells 3 humidifiers models : the Jr. which weighs 10 lbs and comes in an 8 ft3 box; the Momma weighs 20 lbs and comes in an 8 ft3 box; the Big Daddy weighs 60 lbs and comes in a 28 ft3 box. Each delivery van has 248 ft3 of space and can hold a maximum of 440 lbs. In order for the van to be fully loaded how many of each model should it carry.
J= # of Jrs
M=# of Mommas
D=# of Big Daddies

22. 18 13 16 8 2 14 3 4 12 -2 6 J M D J= # of Jrs M=# of Mommas
D=# of Big Daddies 18 13 16 8 2 14 3 4 12 -2 6

23. A doctors prescription calls for a daily intake of liquid containing 40 mg of vitamin C and 30 mg of vitamin D. Your pharmacy stocks 3 liquids in the table below that can be mixed together.
Vitamin C
Vitamin D
Liquid 1
20%
30%
Liquid 2
40%
Liquid 3
50%
total
40 mg
30 mg
How many mg of each liquid should be used? (Assume the solutions need to be whole numbers.)
x = # mg of liquid y = # mg of liquid z = # mg of liquid 3 L1 L2 L3
50mg
75mg
0mg
36mg
76mg
8mg
22mg
77mg
16mg
78mg
24mg

24. Make this problem part of your homework
One group of customers bought 8 deluxe hamburgers, 6 orders of large fries, and 6 colas for $ A second group ordered 10 deluxe hamburgers, 6 large fries, and 8 large colas and paid $ Is there enough info to find the exact cost of each item? Find a solution in terms of z and use that information to fill out the rest of the table.
x = cost of deluxe hamburgers
y = cost of large fries
Z= cost of large colas
Del Ham
L Fries
L Colas
$0.50
$1.25 $2

25. Problem on previous slide P 870 34, 36
Change the riboflavin from 32 to 37 in part a and find 4 possible food combinations that would give these totals.

26. Change the riboflavin from 32 to 37 in part a and find 4 possible food combinations that would give these totals.