gauss elimination with parameters

Posted on 2008-11-19
Medium Priority
Last Modified: 2016-02-10
I'm breaking my head trying to figure out what i'm doing wrong,
i'm support to eliminate the following equations using Gauss elimination system

and i'm suppose to figure out for which a,b (which are Parameters) the system has One solution/Infinit Solutions/No solutions - if there's infint solutions i'm suppose to write the general solution for the system.
now it's all pretty basic in the begining

1    b  1  3
1  2b  1  4 R2->R2-R1
a    1  1  4 R3->R3-aR1

which leads me to

1   b           1  3
0   b           0  1       * i can reach the first value using /:b , i then get y=1/b
0  1-ab   1-a  4-3a

from here i'm puzzled as there's no way to carry on the elimination (if i'm doing it right)
Question by:d3bugg3r
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LVL 27

Expert Comment

ID: 22998098
You are making things too complicated.

What happens if   a = b = 1 ??

What values will make the 1st and 3rd equations wierd??

Author Comment

ID: 22998220
I dont see how it's possible, it will give the following result for the 3rs equation:
0 0 0 0 1 which is not possible, not as a solution anyway.
LVL 27

Expert Comment

ID: 22998258
What happens to the 1st and 2nd equations if   a = b = 1 ??
You don't have to solve them, just look at them.

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Author Comment

ID: 22998415
yes, it works out... but it doesnt for the 3rd so i cannot use those values as answer.
and i have to define if for those two parameters there's uniqe solution, unlimited solutions or no solutions (like if b=0 then 1/0 on the second equation isnt possible)
LVL 27

Accepted Solution

d-glitch earned 500 total points
ID: 22998554
There are three cases to consider:

1st and 2nd    ==>   a = b = 1    ==>  No solutions

1st and 3rd    ==>   a=1  b=0.5  ==>  Infinite solutions

2nd and 3rd   ==>  b=0 (which you found)  ==>  No solutions

I'm pretty sure that all other cases have single solutions.
LVL 22

Assisted Solution

NovaDenizen earned 500 total points
ID: 23005425
Whenever you have an equation involving a product like
r * s = t

When t is zero, at least one of r and s must be zero
When t is nonzero, both r and s must be nonzero

Also, whenever you want to eliminate a factor like in
a*b = 1
to get
a = 1/b
You simply cannot do that unless you have proven that b is nonzero.


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