Finding solutions of linear homogeneous system in integers and rationals.

Question

I encountered a problem given by my linear algebra prof which asks to find integer solutions of $2x+3y+z=0$ .Also there was another problem in which I have to find rational solutions of the system $x+4y+z=0,4x+y-z=0$.I do not know how to get the entire solution set in $\mathbb Q$ or $\mathbb Z$.Please someone help me to solve it. Is the integral solution set of the $1$st one this set $(-3a-b,2a,2b)$ where $a,b$ are integers.And I think the second one consisting of two equations has the solution $(q,-q,2q)$ in the set $\mathbb Q^3$.

Answer 1

The solution of the second system in parametric form is

$$ \begin{pmatrix} x \\ y \\ z \end{pmatrix} = k \cdot \begin{pmatrix} 1 \\ -1 \\ 3 \end{pmatrix}$$

and it suffices to take $k$ rational to find the rational solutions.

The solution of the first system in parametric form is

$$ \begin{pmatrix} x \\ y \\ z \end{pmatrix} = k \cdot \begin{pmatrix} -0.5 \\ 0 \\ 1 \end{pmatrix} + l \cdot \begin{pmatrix} -1.5 \\ 1 \\ 0 \end{pmatrix}.$$

Picking $k$ and $l$ integer is a sufficient and necessary condition for the integrity of the second and third coordinate. However, the sum of $k$ and $l$ must be even in order for the first coordinate to be integer. This yields the solutions

$$ \begin{pmatrix} x \\ y \\ z \end{pmatrix} = k \cdot \begin{pmatrix} -0.5 \\ 0 \\ 1 \end{pmatrix} + (k + 2 \cdot n) \cdot \begin{pmatrix} -1.5 \\ 1 \\ 0 \end{pmatrix}$$

with $k$ and $n$ integer.