Linear algebra systems ,Ax=b,Ax=0

[madah12]

Homework Statement

Let be Ax=b any consistent system of linear equations, and let be x₁ a fixed solution. Show that every solution to the system
can be written in the form x=x₁ +x₀, where x₀ is a solution Ax=0 . Show also that every matrix of this form is a solution

Homework Equations

The Attempt at a Solution

well A(x₁+x₀)= Ax₁+Ax₀=b+0=b
but this only proves that any matrix x=x1+x0 is a solution but not that every solution can be written as such and i don't know how to prove that

 

[HallsofIvy]

Suppose x is a solution to Ax= b. Show that x- x₁ is a solution to Ax= 0.

 

[madah12]

wait x-x0 or x-x1?

 

[madah12]

I mean A(x-x0)=B-0=b isn't a solution but maybe A(x-x1)=b-b = 0 hmm so you are saying there will always exist x0=x-x1 such that x=x1+x0 is a solution to Ax=b

 

[madah12]

I would appreciate a comment about whether i am right or wrong cause i don't want to go to the next problem until I am sure i understood this one