<s>[INST] Determine the conditions on a, b, and c so that the system has a solution? (Matrices) Determine the values of a, b, and c so that the system has a solution:
x + 2y + 4z = a
2x + 3y - z = b
3x + y + 2z = c  
Now, all I did was reduce the system to row-echelon form using gauss elimination as such:
$$ \left[
    \begin{array}{ccc|c}
      1 & 2 & 4 & (a) \\
        2 & 3 & -1 & (b) \\
        3 & 1 & 2 & (c) \\
    \end{array}
\right] $$
~
$$ \left[
    \begin{array}{ccc|c}
      1 & 2 & 4 & (a) \\
        0 & -1 & -9 & (b - 2a) \\
        0 & -5 & -10 & (c - 3a) \\
    \end{array}
\right] $$
~
$$ \left[
    \begin{array}{ccc|c}
      1 & 2 & 4 & (a) \\
        0 & -1 & -9 & (b - 2a) \\
        0 & 0 & 35 & (c +7a -5b) \\
    \end{array}
\right] $$
so the last equation of the system becomes 35 = c + 7a - 5b but from here I'm not sure how to choose the values of a, b, and c for the system to have a solution.
 [/INST] You can further row reduce and end up with:
$$ \left[
    \begin{array}{ccc|c}
      1 & 0 & 0 & \dfrac{1}{5} (2 c-a) \\
 0 & 1 & 0 & \dfrac{1}{35} (7 a+10 b-9 c) \\
 0 & 0 & 1 & \dfrac{1}{35} (7 a-5 b+c) \\
    \end{array}
\right] $$
Now, notice that you have three free variables and they can be chosen any way you like for an infinite number of unique solutions.
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