Solving linear systems + gaussian elimination

[dcramps]

Homework Statement

http://img.rflz.net/problems.jpg

Homework Equations

The Attempt at a Solution

Question 1
x₄ = 1 + 4x₅
but from there I am not sure where to go or if there is even a solution?

Question 2
I eventually got the matrix down to this:

1  2 -3  1   4     1
0  1  1  0   1     1
0  0  1  0  -1/3  -1/3
0  0  0  1  -1/3   4/3

but again I am not sure if that is enough? I don't understand how back substitution is possible.

Question 3
Similar to 3. I sort of got it to row-echelon, but am having trouble. Any help on the previous question should help here.

Thanks

 

[Mark44]

For questions 1 and 2 you have more variables than equations, so you're going to have an infinite number of solutions.

For 1, you can get the system to reduced row-echelon form, which is a step beyond Gaussian elimination. In RREF, each row that has a nonzero first entry has 1 in that position, and all rows above and below have 0 in the same position. Doing this, I was able to get the system in this form:

[1 0 -5 0 36 | 3]
[0 1 1 0 -15 |-3]
[0 0 0 1 -4 | 1]

From this you can write x₁ in terms of x₃ and x₅, x₂ in terms of x₃ and x₅, and x₄ in terms of x₅.

If you do this by back substitution, you'll start with the 3rd equation, and write x₄ in terms of x₅. Substitute for x₄ in the 2nd equation, so you'll eventually get to x₂ in terms of x₃ and x₅. Substitute what you have here for x₂ in the 1st equation to get x₁ in terms of x₃ and x₅. IOW, the same as what I did before using RREF, which is essentially Gauss elimination with back substitution.

For question 2, do the same thing.

 

[dcramps]

Thanks for the reply. I managed to get 2 and 3 on my own, but your advice on question 1 should be very useful. :)

 

[dcramps]

I tried putting the system in the form you have, and I think you have made a mistake. The -3 in the last column should be a -1, no?

 

[Mark44]

Yes, you're right. The last entry in the 2nd row should be -1.

[1 0 -5 0 36 | 3]
[0 1 1 0 -15 |-1]
[0 0 0 1 -4 | 1]