`4x + 3y + 17z = 0, 5x + 4y + 22z = 0, 4x + 2y + 19z =0` Solve the system of linear equations and check any solutions algebraically.

You may subtract the third equation from the first, such that:

`y  - 2z = 0 => y = 2z`

Replace 2z for y in equations 2 and 3, such that:

`5x + 8z + 22z = 0 => 5x+30z = 0 => x + 6z = 0 => x = -6z`

`4x + 4z + 19z = 0 => 4x + 23z = 0`

`4x + 23z - 4x - 24z = 0 => -z = 0 => z = 0 => x = 0 => y = 0`

Hence, evaluating the solution to the homogeneous system, yields x = y = z = 0.