State true or false ?

Question

if $Ax=b$ is consistent, then the solution set of $Ax=b$ is obtained by translating the solution set of $Ax=0$ is it true or false? or is it sometimes false and sometimes true?

Answer 1

HINT: If $Ax=b$ is consistent, it has a solution; let $v$ be a particular solution, so that $Av=b$. Let $S=\{x:Ax=0\}$, and let $T=\{x+v:x\in S\}$.

• Is every vector in $T$ a solution to $Ax=b$?
• Is every solution to $Ax=b$ in $T$?

(Note: this is a basic, standard result that ought to be in just about any elementary linear algebra textbook.)