Question 999480

{{{w+x+y+z=2}}}..................eq.1
{{{2w-x-y+2z=7}}}..................eq.2
{{{2w+3x+2y-z=-2}}}..................eq.3
{{{3w-2x-y-3z=-2 }}}..................eq.4
-----------------------

{{{w+x+y+z=2}}}..................eq.1
{{{2w-x-y+2z=7}}}..................eq.2
------------------add

{{{w+x+y+z+2w-x-y+2z=2+7}}}
{{{3w+3z=9}}}
{{{3(w+z)=9}}}
{{{w+z=3}}}
{{{w=3-z}}}...............1a

{{{2w+3x+2y-z= -2}}}.....................eq.3..multiply by {{{2}}}
{{{3w-2x-y-3z= -2}}} ......................eq.4.multiply by {{{3}}}
-------------------------------------------
{{{4w+6x+4y-2z= -4}}}
{{{9w-6x-3y-9z=-6 }}}
--------------------------------add
{{{4w+6x+4y-2z+9w-6x-3y-9z= -4-6}}}
{{{13w+y-11z= -10}}}

{{{y= 11z-13w-10}}}............substitute {{{ w}}} from 1a

{{{y= 11z-13(3-z)-10}}}

{{{y= 11z-39+13z-10}}}

{{{y= 24z-49}}}.............1b

go to
{{{w+x+y+z=2}}}..................eq.1, subst. {{{w}}} from 1a and {{{y}}} from 1b

{{{(3-z)+x+(24z-49)+z=2}}}

{{{3-z+x+24z-49+z=2}}}
{{{x+24z-46=2}}}
{{{x=46+2-24z}}}
{{{x=48-24z}}}...............1c

go to

{{{2w+3x+2y-z= -2}}}.........eq.3, substitute  {{{w}}} from 1a , {{{y}}} from 1b, and {{{x}}} from 1c

{{{2(3-z)+3(48-24z)+2(24z-49)-z= -2}}}............solve for {{{z}}}

{{{6-2z+144-72z+48z-98-z= -2}}}

{{{-75z+48z+150-98= -2}}}

{{{-27z+52= -2}}}

{{{2+52= 27z}}}

{{{54= 27z}}}

{{{z=54/27}}}

{{{highlight(z=2)}}}

go to 1a substitute {{{z}}}

{{{w=3-z}}}
{{{w=3-2}}}
{{{highlight(w=1)}}}

go to 1b substitute {{{z}}}

{{{y= 24z-49}}}.............1b
{{{y= 24*2-49}}}
{{{y= 48-49}}}
{{{highlight(y= -1)}}}

go to 1c substitute {{{z}}}

{{{x=48-24z}}}...............1c
{{{x=48-24*2}}}
{{{x=48-48}}}
{{{highlight(x=0)}}}

so, your solutions are: 
{{{highlight(x=0)}}}
{{{highlight(y= -1)}}}
{{{highlight(w=1)}}}
{{{highlight(z=2)}}}