$Ax=v$, $Ax=w$ consistent; what about $Ax=v+w$? Let A =
\begin{bmatrix}
    7 & -3 & 5\\
    -4 & 1 & -5 \\
     -5 &2 &-4
  \end{bmatrix}
, v = \begin{bmatrix}
    2 &\\
    1\\
   -1
  \end{bmatrix}, and w = \begin{bmatrix}
    7 &\\
    6\\
   -3
  \end{bmatrix}
, Suppose you know the equations Ax= v and Ax = w are both consistent. What can you say about the equation Ax = v + w?

Ignoring the actual numbers for this question, if $Ax=v$ and $Ax=w$ are consistent, it means both admit solutions (i.e. $Ax_1 = v$ and $Ax_2 = w$.)
Note $A(x_1 + x_2) = Ax_1 + Ax_2 = v + w$ so we have solutions for the equation $Ax=v+w$. i.e. The equation is consistent too.
