`x + 4z = 1, x + y + 10z = 10, 2x - y + 2z = -5` Solve the system of linear equations and check any solutions algebraically.

Eq 1 : `x+4z=1`

Eq 2 : `x+y+10z=10`

Eq 3 :`2x-y+2z=-5`

Multiply equation 1 by -1,

`-1(x+4z)=-1`

`-x-4z=-1`

Now add the above equation and equation 2,

`(-x-4z)+(x+y+10z)=-1+10`

Eq 4:`y+6z=9`

Multiply equation 1 by -2,

`-2(x+4z)=-2`

`-2x-8z=-2`

Now add the above equation and equation 3,

`(-2x-8z)+(2x-y+2z)=-2+(-5)`

Eq 5 : `-y-6z=-7`

Now add the equation 4 and equation 5 i.e.,

`y+6z=9`

`-y-6z=-7`

adding the above equations,

0=2

Hence the equations are inconsistent and have no solution