Suppose that a homogeneous system is consistent and its solu

Suppose that a homogeneous system is consistent and its solution set represents a line through the origin in R^3. What can you say about solutions to a reached nonhomogeneous system?

Solution

We know that the solution set of the equation AX= 0 is a line through the origin when the solution has one free variable.

Further, the solution set of AX = b equals the set S = { w: w = p+ v } where p is a solution of Ax = b and v is a solution of Ax = 0.Hence, the solution set of the equation AX= b is a line not through the origin when the solution has one free variable.