Question 729957
a) 

{{{2x+3y+7z=13}}}........eq. 1                         
{{{3x+2y-5z=-22}}}........eq. 2                          
{{{5x+7y-3z=-28}}}........eq. 3  
________________________

{{{2x+3y+7z=13}}}........eq. 1  ..both multiply: eq1 by 3, eq 2 by 2                         
{{{3x+2y-5z=-22}}}........eq. 2    

{{{6x+9y+21z=39}}}........eq. 1                         
{{{6x+4y-10z=-44}}}........eq. 2  .......subtract
______________________  

{{{6x+9y+21z-(6x+4y-10z)=39-(-44)}}}

{{{6x+9y+21z-6x-4y+10z=39+44}}}

{{{5y+31z=83}}}......eq. 1a


{{{3x+2y-5z=-22}}}........eq. 2        ...both multiply: eq2 by 5, eq 3 by 3                   
{{{5x+7y-3z=-28}}}........eq. 3  
________________________

{{{15x+10y-25z=-110}}}........eq. 2
{{{15x+21y-9z=-84}}}........eq. 3  
________________________.......subtract

{{{15x+10y-25z-(15x+21y-9z)=-110-(-84)}}}

{{{15x+10y-25z-15x-21y+9z)=-110+84}}}

{{{-11y-16z=-26}}}......eq. 2a

take eq. 1a and 2a

{{{5y+31z=83}}}......eq. 1a....both multiply: eq1a by 11, eq 2a by 5

{{{-11y-16z=-26}}}......eq. 2a
_____________________________

{{{55y+341z=913}}}......eq. 1a.
{{{-55y-80z=-130}}}......eq. 2a
_____________________________...add

{{{55y+341z+(-55y-80z)=913+(-130)}}}
{{{55y+341z-55y-80z=913-130}}}

{{{341z-80z=783}}}

{{{261z=783}}}

{{{z=783/261}}}

{{{highlight(z=3)}}}

now go back to {{{55y+341z=913}}}......eq. 1a., plug in {{{z}}} and solve for {{{y}}}

{{{55y+341*3=913}}}

{{{55y+1023=913}}}

{{{55y=913-1023}}}
{{{55y=-110}}}
{{{y=-110/55}}}
{{{highlight(y=-2)}}}


now go back to {{{2x+3y+7z=13}}}........eq. 1., plug in {{{y}}} and {{{z}}} and solve for {{{x}}}

{{{2x+3(-2)+7*3=13}}}........eq. 1

{{{2x-6+21=13}}}

{{{2x+15=13}}}

{{{2x=13-15}}}

{{{2x=-2}}}

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

{{{highlight(x=-1)}}}



b) 
{{{x+3y+5z=20}}}........eq. 1  
{{{y-4z=-16}}}.............eq. 2  
{{{3x+2y+9z=36}}}........eq. 3  
________________________

{{{3x+9y+15z=60}}}........eq. 1  ..multiply: eq1 by 3,                         
{{{3x+2y+9z=36}}}........eq. 3      ...subtract
______________________  

{{{3x+9y+15z-(3x+2y+9z)=60-36}}}
{{{3x+9y+15z-3x-2y-9z=24}}}
{{{7y+6z=24}}}.........................eq. 1a

take
{{{7y+6z=24}}}.........................eq. 1a
{{{y-4z=-16}}}.............eq. 2 ............multiply by {{{7}}} and subtract
______________________________

{{{7y+6z=24}}}.........................eq. 1a
{{{7y-28z=-112}}}.............eq. 2
____________________________subtract
{{{7y+6z-(7y-28z)=24-(-112)}}}

{{{7y+6z-7y+28z=24+112}}}

{{{34z=136}}}

{{{z=136/34}}}

{{{highlight(z=4)}}}

go back to

{{{y-4z=-16}}}.............eq. 2 , plug in {{{z}}} and solve for {{{y}}}

{{{y-4*4=-16}}}

{{{y-16=-16}}}

{{{y=-16+16}}}

{{{highlight(y=0)}}}

go to {{{x+3y+5z=20}}}........eq. 1  plug in {{{y}}} and {{{z}}} and solve for {{{x}}}


{{{x+3*0+5*4=20}}}.

{{{x+0+20=20}}}

{{{x=20-20}}}

{{{highlight(x=0)}}}