Question 1207192


{{{2x-3y-9z=20}}} ....eq.1
{{{x+3z=-2}}} ...........eg.2
{{{-3x+y-4z=-2}}}.....eq.3
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start with

{{{2x-3y-9z=20}}} ....eq.1, solve for {{{x}}}
{{{2x=20+3y+9z }}}
{{{x=10+3y/2+9z/2}}} ....eq.1a

then

{{{x+3z=-2}}} ...........eg.2, solve for{{{ x}}}

{{{x=-2-3z}}} ...........eg.2a


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

{{{10+3y/2+9z/2=-2-3z}}}....solve for{{{ y}}}

{{{3y/2=-2-3z-10-9z/2}}}...both sides multiply by {{{2}}}

{{{3y=-4-6z-20-9z}}}

{{{3y=-15z-24}}}..both sides divide by {{{3}}}

{{{y=-5z-8}}}.....eq.3a


solutions of {{{x }}}and {{{y}}} in terms of {{{z }}}are:

{{{x=-2-3z}}}
{{{y=-5z-8}}}


check:

{{{2x-3y-9z=20}}} ....eq.1, substitute {{{x}}} and{{{ y}}}
{{{2(-2-3z)-3(-5z-8)-9z=20}}} 
{{{-4-6z+15z+24-9z=20}}}
{{{-15z+15z+20=20}}}
{{{20=20}}}=> true


{{{x+3z=-2}}} ...........eg.2, substitute {{{x}}}
{{{-2-3z+3z=-2}}}
{{{-2=-2}}} => true


{{{-3x+y-4z=-2}}}.....eq.3, substitute {{{x}}} and{{{ y}}}
{{{-3(-2-3z)+(-5z-8)-4z=-2}}}
{{{6+9z-5z-8-4z=-2}}}
{{{9z-9z-2=-2}}}
{{{-2=-2}}} =>true