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firat we need to get two equation with two varibles let us work on x,y
so adding the first and last one will yield
now we since we used the first and third we need to use the second to get a correct system.let us multiply the third by 2 then add the second and third
now we have two equation with the variables x and y
to make sure the system works let us check by substituting into the three equations
the first one will be 5x-2y+z=-1--->5(-1)-2(0)+4=-1---->-5+4=-1--->-1=-1 first equation holds
the second equation 3x+y+2z=6---->2(-1)+0+2(4)=6--->-2+8=-6--->-6=-6 second equation holds
the third equation x-3y-z=-5----->-1-3(0)-4=-5---->-1-4=-5--->-5=-5
our third equation also holds which makes our solution correct
x=-1,y=0,z=4