Will $2$ linear equations with $2$ unknowns always have a solution? As I am working on a problem with 3 linear equations with 2 unknowns I discover when I use any two of the equations it seems I always find a solution ok. But when I plug it into the third equation with the same two variables  , the third may or may not cause a contradiction depending if it is a solution and I am OK with that BUT I am confused on when I pick the two equations with two unknowns it seems like it has no choice but to work.  Is there something about linear algebra that makes this so and are there any conditions where it won't be the case that I will find a consistent solution using only the two equations?  My linear algebra is rusty and I am getting up to speed. These are just equations of lines and maybe the geometry would explain it but I am not sure how. Thank you. 

Let's think using vector notation.
A linear system with two unknowns $x$ and $y$, and two equations
$$
  \begin{align*}
    v_1 x + w_1 y &= a_1
    \\
    v_2 x + w_2 y &= a_2
  \end{align*}
$$
can be written in vector notation as
$$
  x\, \vec{v} + y\, \vec{w} = \vec{a}.
$$
That is, you want to know if $\vec{a}$ can be written as a linear combination of $\vec{v}$ and $\vec{w}$.
Fixed the vectors $\vec{v}$ and $\vec{w}$, to state that a solution always exists whatever $\vec{a}$ is, is the same as to state that $\vec{v}$ and $\vec{w}$ spans the whole plane. If not ($\vec{v}$ and $\vec{w}$ are parallel), depending on $\vec{a}$, the solution might not exist. And when it exists, it will not be unique.
