Question

Solve the system using Gaussian elimination. State whether the system is? independent, dependent, or inconsistent.

3x-y+2z=5

x+y-4z=6

Answer 1

the given system of equation can be written in matrix form as

Ax=b where

We reduce the augmented matrix into reduced row echelon form using Gaussian elimination method using elementary row transformation as,

[A|b] =

Here note that rank(A) = rank( [A|b]) = 2 <3

So the system is dependent and it has 3-2=1 linearly independent solution

Since z is a free variable so assume that z=k where k is any real number

Then from the reduced form the system of equation is

x-z/2=11/4 and y-7z/2=13/4

=> x = 11/4+z/2 and y= 13/4+7z/2

Thus the solution set is (x,y,z) = (11/4+k/2, 13/4+7k/2, k) where k is any real number