Find the vector $v$ such that there are an infinite number of solutions to the system of equations of the form $x^{*} + kv$ Q: The system of equations: 
$2x_{1}+x_{2}+3x_{3}=b_{1}$
$x_{2}+x_{3}=b_{2}$
$x_{1}+x_{3}=b_{3}$
b) Find the vector $v$ such that there are an infinite number of solutions to the system of equations of the form $x^{*} + kv$
Understanding:
I've attached a photo of the question with my attempts to the post. 
I've reduced the matrix to echelon form to find solutions for $x_{1}, x_{2}, x_{3}$ and also using the assumption that  $b_{1}= 2b_{3}+b_{2}$, arrived to the confirmation that all 3 equations are independent, and the existance of more free variables than non zero rows leads to the conclusion of infinite solutions. However, I ended up going full circle achieving the equations that were given, and still unsure how to put in the required vector notation.
Revised solution for part A, and full question
Solution for part 2

Hint: Multiplying the third equation by $(-2)$ and adding to the first we get
$$x_2+x_3=b_1-2b_3$$ and the second equation is given by
$$x_2+x_3=b_2$$
Can you proceed?
