Suppose we have a linear system of equations as follows:
$$a_1x+b_1y+c_1z=d_1$$ $$a_2x+b_2y+c_2z=d_2$$ $$a_3x+b_3y+c_3z=d_3$$
The solution to the above system of linear equations is same as that for the following matrix equation:
$$\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix} = \begin{bmatrix}d_1\\d_2\\d_3\end{bmatrix}$$
Is there a specific reason for this equivalence or is it just the way the matrix objects were constructed to behave?
Also how does this specifically relate to vectors?