Solving Systems using Matrices - Inconsistent and Dependent

Systems and Their Applications

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.

Solving Linear SystemsUsing Matrices

Example

3 4 7

4 -2 5

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

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

Gauss-Jordan Elimination

Example

ExampleSolve using Gauss-Jordan Elimination.

Question 1

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

Incorrect!

Check your math, and don’t forget that inconsistent means the system has no solution!

Try again!

Correct!

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

Gaussian Elimination to Systems Without Unique

Solutions

Possible Positions for Three PlanesInconsistent Systems

Dependent Systems

Possible Positions for Three Planes

The Matices for Infinitely Many Solutions and

No Solution

Question 2

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

Incorrect!

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

• Try again.

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!

Nonsquare Systems

Question 3

Solve the previous system using matrices.A. Inconsistent.B. Dependent.

Incorrect!

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

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!

Applications

(a) x + y = 5

(b) x + y = 6

(c) x + y = 11

(d) I1 + x + y = 11

Question 4

Incorrect!

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

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!

(a) (b)

(c) (d)

Question 5

3z = -5

Incorrect!

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

Correct!

Good Job!

(a)

(b)

(c)

(d)

Question 6

Incorrect!

Check your math and try again!

Correct!

Good job!

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?