Question 1059434
{{{R[1]}}}:{{{w + 2x + 2y + z  = -2}}}
{{{R[2]}}}:{{{w + 3x - 2y - z =  -6}}}
{{{R[3]}}}:{{{-2w - x + 3y + 3z =  6}}}
{{{R[4]}}}:{{{ w + 4x + y - 2z =  -14}}}

Eliminate w by using elementary row operations,
{{{R[1]-R[2]}}}:{{{w + 2x + 2y + z -w - 3x + 2y + z =-2-(-6)}}}
{{{R[1]-R[2]}}}:{{{-x+4y+2z= 4}}}
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{{{2R[1]+R[3]}}}:{{{2w+4x+2y+2z-2w-x+3y+3z=-4+6}}}
{{{2R[1]+R[3]}}}:{{{3x+7y+5z=2}}}
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{{{R[1]-R[4]}}}:{{{w + 2x + 2y + z-w-4x-y+2z=-2-(-14)}}}
{{{R[1]-R[4]}}}:{{{-2x+y+3z=12}}}
So now your reformed equations are,
{{{R[5]}}}:{{{-x+4y+2z= 4}}}
{{{R[6]}}}:{{{3x+7y+5z=2}}}
{{{R[7]}}}:{{{-2x+y+3z=12}}}
Continue in this fashion until you have solved for one variable and then work backwards. 
You can eliminate {{{x}}} by these row operations,
{{{3R[5]+R[6]}}} and
{{{-2R[5]+R[7]}}}