When working out this system of equations, I've found that there are no solutions.
Is there any way from the start you can Identify that this system of equations has no solutions?
$2x + y − z + u = 1$
$3x − 2y + 2z − 3u = 2$
$5x + y − z + 2u = −1$
$2x − y + z − 3u = 4$
$\endgroup$ 51 Answer
$\begingroup$The only way is to form the matrix of the system and do row reductions: $$ \left[ \begin{array}{cccc|c} 2&1&-1&1&1\\3&-2&2&-3&2\\5&1&-1&2&-1\\2&-1&1&-3&4 \end{array}
\right] $$
For fewer variables, sometimes you can try to graph them out to see whether the lines are intersecting, same, or parallel, and from there you can decide if the system if consistent. But in this case where there are four variables, obtaining the row-echelon form would probably be the fastest way.
