Final answer:
To find the general solution for the system of linear equations using Gauss-Jordan elimination, we first write the augmented matrix and perform row operations to obtain row-echelon form. An example of a non-zero solution is provided. The equation of the plane passing through given points is also found.
Step-by-step explanation:
To solve the given system of linear equations using Gauss-Jordan elimination, we first write the augmented matrix:
. 1 -1 -1 1 0
. 0 1 3 -2 0
. 2 4 4 -2 0
Perform row operations to transform the augmented matrix into row-echelon form:
. 1 -1 -1 1 0
. 0 1 3 -2 0
. 0 0 0 0 0
From the row-echelon form, we can form the system of equations as:
. z1 - z2 - z3 + z4 = 0
. z2 + 3z3 - 2z4 = 0
Since we have a free variable (z4), we can express the solutions in terms of it.
An example of a non-zero solution is: z1 = 1, z2 = -1, z3 = 0, z4 = 1.
The points (1, 0, 3), (1, 1, 1), and (-2, -1, 2) lie on the plane defined by the equation 2x - y + z = 3.