Question 148619
1.{{{4x+3y+2z=6}}}
2.{{{-2x-y+5z=5}}}
3.{{{x+2y-3z=3}}}
Combine equations to remove one of the variables. 
Multiply eq. 2 by (3) and add to eq. 1,
2.{{{3(-2x-y+5z)=3(5)}}}
2.{{{-6x-3y+15z=15}}}
This gets rid of y and makes a new eq. 1 with just x and z,
1.{{{4x+3y+2z-6x-3y+15z=6+15}}}
1.{{{-2x+17z=21}}}
Multiply eq. 2 by (2) and add to eq. 3,
2.{{{2(-2x-y+5z)=2(5)}}}
2.{{{-4x-2y+10z)=10}}}
This gets rid of y and makes a new eq. 3 with just x and z,
3.{{{x+2y-3z-4x-2y+10z=3+10}}}
3.{{{-3x+7z=13}}}
Now we have two equations in x and z.
1.{{{-2x+17z=21}}}
3.{{{-3x+7z=13}}}
We can continue to reduce to one variable. 
Multiply eq. 1 by 3 and eq. 3 by -2 and add them to get rid of x,
1.{{{3(-2x+17z)=3(21)}}}
1.{{{-6x+51z=63}}}
3.{{{-2(-3x+7z)=-2(13)}}}
3.{{{6x-14z=-26}}}
Now add the two eqs.
{{{-6x+51z+6x-14z=63-26}}}
{{{37z=37}}}
{{{z=1}}}
Now that you have z, work backwards and back substitute to find x and then y.
You can use any of the previous equations to solve for other variables.
{{{-3x+7z=13}}}
{{{-3x+7(1)=13}}}
{{{-3x+7=13}}}
{{{-3x=6}}}
{{{x=-2}}}
And finally for y,
2.{{{-2x-y+5z=5}}}
{{{-2(-2)-y+5(1)=5}}}
{{{4-y+5=5}}}
{{{y=4}}}
(x,y,z)=(-2,4,1)