Consider the given equations:
[tex]-x+2y-5z = 19[/tex] (Equation 1)
[tex]6x-3y-z= -4[/tex] (Equation 2)
[tex]x-4y-z=21[/tex] (Equation 3)
Adding equations 1 and 3, we get
[tex]-x+2y-5z+x-4y-z = 19+21[/tex]
[tex]-2y-6z = 40[/tex]
So, we get [tex]-y -3z = 20[/tex] (Equation 4)
Multiplying equation 3 by '6', we get
[tex]6x - 24y-6z = 126[/tex] (Equation 5)
Subtracting equation 5 from equation 2, we get
[tex](6x-3y-z)-(6x - 24y-6z) = -4-126[/tex]
[tex]6x-3y-z-6x + 24y+6z) = -4-126[/tex]
[tex]21y+5z = -130[/tex] (Equation 6)
Multiplying equation 4 by '21' and adding it to equation 6, we get
[tex]-21y-63z+21y+5z = 420-130[/tex]
[tex]-58z = 290[/tex]
So, z = -5
Since, [tex]-y-3z = 20[/tex] [tex]-y+15=20[/tex] [tex]y=-5[/tex]
So, y=-5
Now, [tex]x-4y-z=21[/tex] [tex]x-4(-5)-(-5)=21[/tex] [tex]x+20+5=21[/tex] [tex]x+25=21[/tex] [tex]x= -4[/tex]
So, x = -4
Therefore, x = -4, y= -5 and z= -5 are the solutions to the given equations.
