Question 1175559

{{{-4x+5y-3z=17}}}.....eq.1
{{{-3x-2y-4z=-1}}}.....eq.2
{{{5x+5y+4z=12}}}.....eq.3
----------------------------------------

start with
{{{5x+5y+4z=12}}}.....eq.3
{{{-4x+5y-3z=17}}}.....eq.1
-------------------------------------------subtract eq.1 from eq.3

{{{5x+5y+4z-(-4x+5y-3z)=12-17}}}.....eq.3
{{{5x+5y+4z+4x-5y+3z=-5}}}
{{{9x+7z=-5}}}....solve for {{{x}}}
{{{9x=-7z-5}}}
{{{x=-7z/9-5/9}}}.....eq.1a


go to

{{{-3x-2y-4z=-1}}}.....eq.2
{{{5x+5y+4z=12}}}.....eq.3
-------------------------------------------add eq.2 from eq.3

{{{5x+5y+4z+(-3x-2y-4z)=12+(-1)}}}
{{{5x+5y+4z-3x-2y-4z=11}}}
{{{2x+3y=11}}}...solve for {{{x}}}
{{{2x=11-3y}}}
{{{x=11/2-3y/2}}}.....eq.1b


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

{{{-7z/9-5/9=11/2-3y/2}}}..solve for {{{y}}}
{{{3y/2=11/2+7z/9+5/9}}}
{{{3y/2=(7 z)/9 + 109/18}}}
{{{y=(2/3)(7 z)/9 + (2/3)109/18}}}
{{{y=(14 z)/27 + 109/27}}}.....eq.1c


go to

{{{x=11/2-3y/2}}}.....eq.1b, substitute {{{y}}}
{{{x=11/2-3((14 z)/27 + 109/27)/2}}}
{{{x=11/2-(3/2)((14 z)/27 + 109/27)}}}..........1d 


{{{-3x-2y-4z=-1}}}.....eq.2, substitute {{{x}}} and {{{y}}} from 1d and 1c

{{{-3(11/2-(3/2)((14 z)/27 + 109/27))-2((14 z)/27 + 109/27)-4z=-1}}}

{{{-1/27 (73 z + 173)=-1}}}
{{{-1 (73 z + 173)=-27}}}
{{{-73 z - 173=-27}}}
{{{27- 173=73 z}}}
{{{-146=73 z}}}
{{{z=-146/73 }}}
{{{z=-2 }}}


go to

{{{y=(14 z)/27 + 109/27}}}.....eq.1c, substitute {{{z}}}
{{{y=(14(-2))/27 + 109/27}}}
{{{y=-28/27 + 109/27}}}
{{{y=81/27}}}
{{{y=3}}}


go to

{{{x=11/2-3y/2}}}.....eq.1b, substitute {{{y}}}

{{{x=11/2-(3*3)/2}}}
{{{x=11/2-9/2}}}
{{{x=2/2}}}
{{{x=1}}}


solution set:
{{{x=1}}}
{{{y=3}}}
{{{z=-2 }}}