Finding system of linear equations starting from parametric solution I need to find a system of two linear equations in variables $x_1$, $x_2$ and $x_3$ from a solution vector of the form $x_1=t$, $x_2=1+t$ and $x_3=2-t$, but I'm not sure where to start.

Each individual implicit equation $ax_1+bx_2+cx_3+d=0$ represents a plane with normal $\mathbf n = (a,b,c)^T$. A line $t\mathbf v+\mathbf p_0$ that lies on this plane is perpendicular to $\mathbf n$, i.e., $\mathbf n\cdot\mathbf v=0$, which in this case produces the constraint $a+b-c=0$. Obviously, any point on the line must lie on the plane, too. We know that $(0,1,2)^T$ is on the line, which generates the equation $b+2c+d=0$. Any two independent solutions to this system of equations will give you the pair of equations that you seek. To put it slightly differently, any two linearly independent elements of the null space of $$\begin{bmatrix}1&1&-1&0\\0&1&2&1\end{bmatrix}$$ give the coefficients of the two required equations.  
If you interpret the rows of this matrix as homogeneous coordinates of points, the first row is the “point at infinity” that corresponds to the direction of the line, and the second is the known point on the line from its definition. This method works in general: if you know any two points on the line (finite or not), assemble them into a matrix and find its null space. If the point is finite, append a $1$ to that row; if the point is at infinity (i.e., is a direction vector for the line), append a $0$.
