Question 1196448

{{{2w + x - y + 2z = 10}}} ..............eq.1
{{{-w - 3x + y - 5z = -11}}}..............eq.2
{{{3w + 5x - 3y + z = 35}}}..............eq.3
{{{-4w - 2x - 3y - z = 2}}}..............eq.4


start with

{{{2w + x - y + 2z = 10 }}}..............eq.1
{{{-w - 3x + y - 5z = -11}}}..............eq.2
_____________________________________add both to eliminate {{{y}}}

{{{2w + x - y + 2z -w - 3x + y - 5z = 10 -11}}}
{{{-3 z - 2 x + w= -1}}}............solve for {{{w}}}
{{{w= -1+3 z +2x }}}........................eq.1a


now go with

{{{3w + 5x - 3y + z = 35}}}..............eq.3
{{{-4w - 2x - 3y - z = 2}}}..............eq.4
___________________________________subtract  eq.4 from eq.3

{{{3w + 5x - 3y + z-(-4w - 2x - 3y - z) = 35-2}}}
 {{{7w+ 7x+2z  =33}}}..........solve for {{{w}}}
{{{7w  =33-7x-2z}}}
{{{w  =33/7-x-2z/7}}}..............eq.2a


from eq.1a and eq.2a we have


{{{-1+3 z +2x=33/7-x-2z/7}}}............solve for{{{ z}}}
{{{2z/7+3 z =33/7-x+1-2x}}}
{{{(23/7) z=40/7 - 3 x}}}
{{{z=(40/7 - 3 x)/(23/7)}}}
{{{z=40/23 - (21/23)x}}}..............eq.3a 


go to 

{{{w= -1+3 z +2x}}} ........................eq.1a, substitute {{{z }}}
{{{w= -1+3 (40/23 - (21/23)x) +2x }}} 
{{{w= 97/23 - (17/23 )x}}}..........eq.4a 


go to

{{{-w - 3x + y - 5z = -11}}}..............eq.2 , substitute {{{w}}} and {{{z}}} 
{{{-(97/23 - (17/23 )x) - 3x + y - 5(40/23 - (21/23) x) = -11}}}

{{{(53/23) x+ y - 297/23= -11}}}................solve for {{{y}}}

{{{y=-(53/23) x+ 297/23-11}}}

{{{y= -(53/23)x+44/23}}}........................eq.5a 



go to

{{{-4w - 2x - 3y - z = 2}}}..............eq.4, substitute {{{w}}}, {{{y}}} and{{{ z}}}


{{{-4(97/23 - (17/23 )x) -2x - 3(- (53/23)x+44/23) -(40/23 - (21/23) x) = 2}}}..............solve for {{{x}}}

{{{(202 /23)x - 606/23 = 0}}}
{{{(202 /23)x = 606/23  }}}
{{{x=(606/23 )/(202 /23)}}}
{{{x=606/202}}}
{{{x=3}}}


now we can find {{{w}}},{{{y}}}, and{{{ z}}}


{{{w= 97/23 - (17/23 )x}}}..........eq.4a , substitute{{{ x}}}
{{{w= 97/23 - (17/23 )3 }}}
{{{w= 97/23 - 51/23}}} 
{{{w= 46/23 }}}
{{{w= 2}}}


 {{{y= - (53/23)x+44/23}}}........................eq.5a , substitute{{{ x}}}
{{{y= - (159/23)+44/23}}}
{{{y= - 115/23 }}}
{{{y=-5}}}


{{{z=40/23 - (21/23)x}}}..............eq.3a , substitute {{{x}}}
{{{z=40/23 - (21/23)3}}}
{{{z=40/23 - (63/23)}}}
{{{z= -(23/23)}}}
{{{z=-1}}}


solutions:
 {{{x = 3}}}
{{{y = -5}}}
{{{w = 2}}}
{{{z = -1}}}