Solve the following problems Be sure to check with your labr

Solve the following problems. Be sure to check with your lab/recitation instructor about what is required. Will you have to turn in a project report? If so, when will it be due? Is there a required format for the report? Be clear on what is expected before your group starts to work. This project will help reinforce some key ideas related to solving linear systems. Answer the following questions about the linear system {x +cz = 6 3x - 2y + z = 2 2x -y + z = 4 1. Set c = -1 and find the unique solution to this system of equations. 2. Find the value of c that corresponds to an infinite number of solutions to this system of equations. 3. Describe the family of solutions given by the value of c in Problem 2. 4. Is there a value of c for which this system has no solutions? Explain why or why not.

Solution

1. c=-1

Denote first equation by A, second by B and third by C

2C-B gives:

x+z=6

From A we have:x-z=6

Hence, x=6,z=0

Substituting this in last equation gives:

12-y=4

y=8

2.

As we saw before: 2C-B gives:x+z=6

First equation is:

x+cz=6

So setting c=1 gives us same as:2C-B

ie linearly dependent equations. In this case we get infinitely many solutions.

3.

x+z=6 hence, x=6-z

2x-y+z=4

2(6-z)-y+z=4

12-y-z=4

y=8-z,x=6-z

This is the required solution.

(6-z,8-z,z)

4.

x+z=6

x+cz=6

Hence, (c-1)z=0

So for c not equal to 1, z=0,x=6 and y=8

Hence for all values of c we have a solution.