c) \( \left\{\begin{array}{l}S_{2}-S_{4}=14-x_{2} \\ x_{3}+S_{3}=10\end{array} ;\right. \) d) \( \left\{\begin{array}{l}x_{1}^{2}+x_{4}^{2}=101 \\ x_{2}+x_{6}=20\end{array}\right. \); e) \( \left\{\begin{array}{l}x_{1}^{2}+x_{3}^{2}=50 \\ S_{3}=12\end{array} ;\right. \) f) \( \left\{\begin{array}{l}x_{2}+x_{4}=16 \\ x_{1} x_{5}=28\end{array}\right. \) The arithmetic progress of the following cases

c) From the first equation, we have $S_2 - S_4 = 14 - x_2$. Rearranging, we get $S_2 = 14 - x_2 + S_4$. Substituting this into the second equation, we have $x_3 + S_3 = 10$. Rearranging, we get $S_3 = 10 - x_3$. Therefore, the system of equations can be rewritten as: \[ \left\{ \begin{array}{l} S_2 = 14 - x_2 + S_4 \\ S_3 = 10 - x_3 \end{array} \right. \] d) From the first equation, we have $x_1^2 + x_4^2 = 101$. Rearranging, we get $x_1^2 = 101 - x_4^2$. Substituting this into the second equation, we have $x_2 + x_6 = 20$. Therefore, the system of equations can be rewritten as: \[ \left\{ \begin{array}{l} x_1^2 = 101 - x_4^2 \\ x_2 + x_6 = 20 \end{array} \right. \] e) From the first equation, we have $x_1^2 + x_3^2 = 50$. Rearranging, we get $x_1^2 = 50 - x_3^2$. Substituting this into the second equation, we have $S_3 = 12$. Therefore, the system of equations can be rewritten as: \[ \left\{ \begin{array}{l} x_1^2 = 50 - x_3^2 \\ S_3 = 12 \end{array} \right. \] f) The system of equations is already in its simplest form.