I have the following system of equations where the m's are known but $a,b,c,x,y,z$ are unknown. How does one go about solving this system? All the usual linear algebra tricks I know don't apply and I don't want to do it through tedious substitutions.

$\begin{array}{rl}a+b+c& ={m}_{0}\\ ax+by+cz& ={m}_{1}\\ a{x}^{2}+b{y}^{2}+c{z}^{2}& ={m}_{2}\\ a{x}^{3}+b{y}^{3}+c{z}^{3}& ={m}_{3}\\ a{x}^{4}+b{y}^{4}+c{z}^{4}& ={m}_{4}\\ a{x}^{5}+b{y}^{5}+c{z}^{5}& ={m}_{5}\end{array}$

$\begin{array}{rl}a+b+c& ={m}_{0}\\ ax+by+cz& ={m}_{1}\\ a{x}^{2}+b{y}^{2}+c{z}^{2}& ={m}_{2}\\ a{x}^{3}+b{y}^{3}+c{z}^{3}& ={m}_{3}\\ a{x}^{4}+b{y}^{4}+c{z}^{4}& ={m}_{4}\\ a{x}^{5}+b{y}^{5}+c{z}^{5}& ={m}_{5}\end{array}$