TITLE: Gaussian elimination with 4 variables
QUESTION [0 upvotes]: After this step however, I end up with 2 simultanious equations but still with four unknowns, I'm not sure if my gaussian elimination is wrong or not but how will i find the exact values of x,y,z and w?

REPLY [1 votes]: If the solution you reach is correct, then, $y$ and $w$ can take any value, and $x$ and $z$ are equal to $x=100-3y+96w$ and $z=54+52w$.
$y$ and $w$ can take any value because the equations 3 and 4 are equivalent to: $0x+0y+0z+0w=0$ and from here, because those equations are pivotal: $0z=0$ and $0w=0$.
As you see, because the gaussian elimination discarded 2 equations, we have 4 variables and 2 LI equations, thus the space of available solutions has dimension 4-2=2. 
$y$ and $w$ are the 2 free variables parametrizing that solution space.
Actually, you can say that, i.e. $x$ and $w$ are your 2 free variables. The problem is the same in this case, if you reinterpret the problem by swapping the columns and proceeding as usually.
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