In linear algebra. Just learned about linear combinations, and solved the following using gaussian elimination: $$c1\begin{bmatrix} 1 \\\ -2 \\\ -5 \end{bmatrix} + c2\begin{bmatrix} 2 \\\ 5 \\\ 6 \end{bmatrix} = \begin{bmatrix}7\\\ 4\\\ -3 \end{bmatrix}$$ This case made total sense to me. We created an augmented matrix and solved.
However, I was thinking what if our combo had been longer? What if there were 4 scalars that we were trying to solve for, but the vectors still were only 3 dimensional? Is something like this possible? $$c1\begin{bmatrix} 1 \\\ -2 \\\ -5 \end{bmatrix} + c2\begin{bmatrix} 2 \\\ 5 \\\ 6 \end{bmatrix} + c3 \begin{bmatrix}7\\\ 4\\\ -3 \end{bmatrix} + c4 \begin{bmatrix}7\\\ 4\\\ -3 \end{bmatrix} = \begin{bmatrix}randomint\\\ randomint\\\ randomint \end{bmatrix}$$
In my feeble understanding, this is not possible. Am I missing something?