1. Solving Systems using Matrices
- Inconsistent and Dependent
Systems and Their Applications

2. Throughout this lesson there will be a series
of questions based on what you have just
learned. Please work towards the solution in
your notes and make a notation based on
whether you answer each question correctly
or not. There will be further directions at the
end of the presentation.
Please raise your hand if you have any
questions or need me to expand on the
explanation given at any point within the
lesson.

3. Solving Linear Systems
Using Matrices

5. Example
3 4 7
4 -2 5

6. We use row operations on the augmented
matrix. These row operations are just like
what we did when using elimination.

7. Example
The new row two will be replaced with
four times row one plus one of row two
Complete the row operation R2 = 4r1 + r2

8. Gauss-Jordan Elimination

11. Example

12. Example
Solve using Gauss-Jordan Elimination.

13. Question 1
A. (3, 4/3, 5)
B. (31/2, 42/3, 47/6)
C. (2, -2, 1)
D. Inconsistent

14. 14
Incorrect!
Check your math, and don’t forget
that inconsistent means the
system has no solution!
Try again!

15. 15
Correct!
Good job!
If you had any problems finding
this solution look through the
solution.

17. Gaussian Elimination to
Systems Without Unique
Solutions

18. Possible Positions for Three Planes
Inconsistent Systems
Dependent Systems

19. Possible Positions for Three Planes

20. The Matices for Infinitely Many Solutions and
No Solution

21. Question 2
1. Solve the above system using matrices.
A. Inconsistent.
B. Dependent.
C. (3, 19, 8).
D. (1, 11, 3).

22. Incorrect!
• Check your math again and also
remember what an inconsistent
and dependent solution look like
in a matrix.
• Try again.

23. Correct!
• Because during the solving process you
run into situation where you have an
entire row of zero’s equaling a non-zero
answer this system is inconsistent. If
you had trouble finding this answer look
through the solution.
• Good job!

25. Nonsquare Systems

27. Question 3
Solve the previous system using matrices.
A. Inconsistent.
B. Dependent.

28. 28
Incorrect!
Check your math again and think
about the likelihood that this matrix
would not be a dependent matrix?
Try again.

29. 29
Correct!
This system can not have a singular
solution and so must have a dependent
solution if not inconsistent. If you had
trouble finding the solution look here.
Good job!

31. Applications

33. (a) x + y = 5
(b) x + y = 6
(c) x + y = 11
(d) I1 + x + y = 11
Question 4

34. 34
Incorrect!
Think about the total number of
cars crossing through the
intersection and write an equation
based on that number
combination.
Try again.

35. 35
Correct!
Yes, because the first road has five cars
going through the intersection and the
second has six we then add them
together and have the equation that
equals eleven.
Good job!

36. (a) (b)
(c) (d)
Question 5
3z = -5

37. 37
Incorrect!
Check your math and remember
that you are replacing row three!
Try again.

38. 38
Correct!
Good Job!

39. (a)
(b)
(c)
(d)
Question 6

40. 40
Incorrect!
Check your math and try again!

41. 41
Correct!
Good job!

42. This is the end of the presentation, please make
sure to reflect on your answers to the five
questions throughout the lesson.
• If you answered any incorrectly the first time please
reflect on why that was?
• Did you make a simple mathematical error?
• Was your understanding of the question or topic not
enough to help guide you to the correct solution and
approach?
• Did choosing the incorrect answer and reading the
hint given help you to correctly solve it?