Question 1193249

{{{x-2y+3z=7}}}........eq.1
{{{2x+y+z=4}}}........eq.2
{{{-3x+2y-2z=-10}}}........eq.3
_____________________


{{{x-2y+3z=7}}}........eq.1.......multiply by {{{2}}}
{{{2x+y+z=4}}}........eq.2
_________________then subtract eq.2 from eq.1


{{{2x-4y+6z-(2x+y+z)=14-4}}}

{{{2x-4y+6z-2x-y-z=10}}}

{{{-5y+5z=10}}}.........divide by {{{5}}}

{{{-y+z=2}}}...........solve for {{{z}}}

{{{z=y+2}}}..........eq.1a



{{{2x+y+z=4}}}........eq.2.......multiply by {{{2}}}
{{{-3x+2y-2z=-10}}}........eq.3
_________________then subtract eq.3 from eq.2

{{{4x+2y+2z-(-3x+2y-2z)=8-(-10)}}}

{{{4x+2y+2z+3x-2y+2z=8+10}}}

{{{7x+4z=18}}}........solve for {{{z}}}

{{{4z=-7x+18}}}

{{{z=-7x/4+18/4}}}

{{{z=-7x/4+9/2}}}..........eq.2a


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


{{{y+2=-7x/4+9/2}}}.......solve for {{{y}}}

{{{y=-7x/4+9/2-2}}}

{{{y=-7x/4+5/2}}}........eq.3a



go to


{{{x-2y+3z=7}}}........eq.1, substitute {{{y}}} and {{{z}}} from eq.3a  and eq.2a

{{{x-2(-7x/4+5/2)+3(-7x/4+9/2)=7}}}.........solve for {{{x}}}

{{{x+7x/2-10/2-21x/4+27/2=7}}}

{{{x+7x/2-10/2-21x/4+27/2=7}}}..........multiply by {{{4}}} to get a rid of fractions

{{{4x+14x-20-21x+54=28}}}

{{{-3x=28+20-54}}}

{{{-3x=-6}}}

{{{x=2}}}


go to

{{{y=-7x/4+5/2}}}........eq.3a, substitute {{{x}}}

{{{y=-(7*2)/4+5/2}}}
 
{{{y=-7/2+5/2}}} 

{{{y=-2/2}}} 

{{{y=-1}}} 


go to


{{{z=y+2}}}..........eq.1a, substitute {{{y}}}

{{{z=-1+2}}} 

{{{z=1}}} 

solutions:
{{{x=2}}}
{{{y=-1}}} 
{{{z=1}}}