- #1
Kolika28
- 146
- 28
- Homework Statement
- I need help with this problem. I have tried everything and get different answers all the time. Given the three equations under, I'm supposed to find out for which value of k do we have
a) No soultion b) A unique solution c) Infinitely many solutions
- Relevant Equations
- My teacher has told me the following is true if the matrix is in row echelon form :
1. If there are one or more columns with no pivot-number, there are infinitely many solutions
2. If every column contains a pivot number, its a unique solution.
3. If every element in the row is zero, except for the argument column, there is no solution.
The first thing I do is making the argumented matrix:
Then I try to rearrange to make the row echelon form. But maybe that's what confusses me the most. I have tried different ways of doing it, for example changing the order of the equations. I always end up with ##k+number## expression in the second column in the third row that I try to make zero. I have tried to different ways to make the row echelon form now:
1.
Rearrange R1 and R3:
Then taking -2R1+R2 and -R1+R3
The last thing I did was multiplying R3 with ##\frac{1}{k-10}## and afterwards took ##\frac{1}{21}##R2+R3:
From rule number 3 there must be no solution when k=-5 because then row 3 will be 0 - 0 - 0 - (8/105). And if k=3, the last row will be 0 - 0 - 0 - 0. From rule number one, there will be infinite many solutions.
2. If I use this calculator: Calculator, I get that the row echelon form is:
This is a very different matrix then what I calculated. But if I assume that this is the right answer and given the rules my teacher has taught me, there will be no solution when k=-5 because then the last row will be 0 - 0 - 0 - (2/3). And if k=3, the last row will be 0 - 0 - 0 - 0. From rule number one, there will be infinite many solutions.So with both row echelon forms of the matrix, I find the numbers -5 and 3. But I don't manage to find out for which k we have a unique solution. What confuses me even more are the fractions I get containing k in both examples. k=10 is not valid in example 1 and k=-1/2 is not valid in example 2. Why are these values different in both examples? A friend of me told me that she got 1/2 as the unique solution, but I don't see how. Thank you if you have taken the time reading everything. I really need some help with this one, because I have been stuck on this problem for three days now.