Question 907725
OK so use row operations and elimination to reduce your variables.
.
.
{{{R[1]}}}:{{{x+y+z-w=5}}}
{{{R[2]}}}:{{{2x+y-z+w=6}}}
{{{R[3]}}}:{{{x-5y+3z+w=9}}}
{{{R[4]}}}:{{{-x-y+z+4w=7}}} 
Let's get rid of {{{w}}} first.
{{{R[5]=R[1]+R[2]}}}: {{{x+y+z-w+2x+y-z+w=5+6}}}
{{{R[5]=R[1]+R[2]}}}: {{{3x+2y=11}}}
.
.
{{{R[6]=R[1]+R[3]}}}: {{{x+y+z-w+x-5y+3z+w=5+9}}}
{{{R[6]=R[1]+R[3]}}}: {{{2x-4y+4z=14}}}
.
.
{{{R[7]=4*R[1]+R[4]}}}: {{{4x+4y+4z-4w-x-y+z+4w=20+7}}}
{{{R[7]=4*R[1]+R[4]}}}: {{{3x+3y+5z=27}}}
.
.
OK, {{{R[5]}}} already has {{{z}}} eliminated, so let's pair up {{{R[6]}}} and {{{R[7]}}} to do the same.
{{{R[8]=5*R[6]-4*R[7]}}}:{{{10x-20y+20z-12x-12y-20z=70-108}}}
{{{R[8]=5*R[6]-4*R[7]}}}:{{{-2x-32y=-38}}}
.
.
Finally, multiply {{{R[5]}}} by {{{16}}} and add to {{{R[8]}}} to eliminate {{{y}}}.
{{{R[9]=16*R5+R8}}} : {{{48x+32y-2x-32y=176-38}}}
{{{R[9]=16*R5+R8}}} : {{{46x=148}}}
So then,
{{{x=148/46}}}
{{{x=3}}}
Then work backwards,
{{{R[5]}}}: {{{3(3)+2y=11}}}
{{{2y=2}}}
{{{y=1}}}
.
.
{{{R[6]}}}: {{{2(3)-4(1)+4z=14}}}
{{{6-4+4z=14}}}
{{{4z=12}}}
{{{z=3}}}
.
.
.
{{{R[1]}}}:{{{3+1+3-w=5}}}
{{{w=2}}}