Question 995315
{{{6x-y+2z=8}}}.....eq.1
{{{2x+3y-z=-9}}}.....eq.2
{{{4x+2y+5z=1}}}.....eq.3
------------------------

here you have the system of three equations with three unknown variables: {{{x}}},{{{y}}}, and {{{z}}}

first, take two equations and try to eliminate one unknown variable


{{{6x-y+2z=8}}}.....eq.1
{{{2x+3y-z=-9}}}.....eq.2....both sides multiply by {{{2}}}
-----------------------------

{{{6x-y+2z=8}}}.....eq.1
{{{4x+6y-2z=-18}}}.....eq.2
------------------------------add both

{{{6x-y+2z+4x+6y-2z=8+(-18)}}}

{{{10x+cross(2z)+5y-cross(2z)=8-18}}}

{{{10x+5y=-10}}}...solve for {{{x}}}

{{{10x=-5y-10}}}

{{{x=-5y/10-10/10}}}

{{{x=-y/2-1}}}................eq.1a

do same with other two equations:

{{{2x+3y-z=-9}}}.....eq.2..........both sides multiply by {{{5}}}
{{{4x+2y+5z=1}}}.....eq.3
-----------------------

{{{10x+15y-5z=-45}}}.....eq.2
{{{4x+2y+5z=1}}}.....eq.3
-----------------------------add

{{{10x+15y-cross(5z)+4x+2y+cross(5z)=-45+1}}}

{{{14x+17y=-44}}}

{{{14x=-17y-44}}}

{{{x=-17y/14-44/14}}}

{{{x=-(17/14)y-22/7}}} ................eq.1b

since 1a and 1b have same left sides, equal the right sides and solve for {{{y}}}

{{{-y/2-1=-(17/14)y-22/7}}}

{{{(17/14)y-y/2=1-22/7}}}

{{{17y/14-7y/14=7/7-22/7}}}

{{{10y/14=-15/7}}}

{{{10y*7=-15*14}}}

{{{70y=-210}}}

{{{y=-210/70}}}

{{{highlight(y=-3)}}}

go back to 1a, plug in {{{-3}}} for {{{y}}} and solve for {{{x}}}

{{{x=-y/2-1}}}................eq.1a

{{{x=-(-3)/2-1}}}

{{{x=3/2-1}}}

{{{x=3/2-2/2}}}

{{{highlight(x=1/2)}}}


now go to any of given eq. plug in values for {{{x}}} and {{{y}}}, and solve for {{{z}}}

{{{2x+3y-z=-9}}}.....eq.2

{{{2(1/2)+3(-3)+9=z}}}

{{{1-cross(9)+cross(9)=z}}}

{{{highlight(z=1)}}}


so, your solutions are:

{{{highlight(x=1/2)}}}

{{{highlight(y=-3)}}}

{{{highlight(z=1)}}}