`5x - 3y + 2z = 3, 2x + 4y - z = 2, x + y - z = -1` Solve the system of linear equations and check any solutions algebraically.

Eq. 1 : `5x-3y+2z=3`

Eq. 2 : `2x+4y-z=2`

Eq. 3 : `x+y-z=-1`

Multiply Eq. 2 by 2 and add Eq. 1,

`4x+8y-2z=4`

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

Eq.4 : `9x+5y=7`

Subtract Eq. 3 from Eq. 2,

Eq.5 : `x+3y=3`

Now let's solve Eq.4 and Eq.5 by substitution method,

From Eq.5 ,

`x=3-3y`

Substitute the above expression of x in the Eq.4,

`9(3-3y)+5y=7`

`27-27y+5y=7`

`27-22y=7`

`-22y=7-27`

`-22y=-20`

`y=(-20)/-22`

`y=10/11`

Plug in the value of y in the expression of x,

`x=3-3y`

`x=3-3(10/11)`

`x=3-30/11`

`x=(33-30)/11`

`x=3/11`

Plug in the values of x and y in the Eq.3,

`3/11+10/11-z=-1`

`13/11-z=-1`

`-z=-1-13/11`

`-z=(-11-13)/11`

`-z=(-24)/11`

`z=24/11`

Solutions of the equations are x=3/11 , y=10/11 and z=24/11