Find the ordered quintuplet that satisfies the system of equations

We can write the equation as

$$\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 4 & 5 & 1 \\ 3 & 4 & 5 & 1 & 2 \\ 4 & 5 & 1 & 2 & 3 \\ 5 & 1 & 2 & 3 & 4 \end{pmatrix} \begin{pmatrix} a \\ b \\ c \\ d \\ e \end{pmatrix} = \begin{pmatrix} 47 \\ 15 \\ 3 \\ 26 \\ 29 \end{pmatrix} .$$
 

Inverting the matrix, we get the solution (a,b,c,d,e) = (-5, 4, 14, 3, -2).