I have a system of equations and I would like your help to show that it has a unique solution with respect to $\lambda_1,\lambda_2,\lambda_3$.
More precisely, let the system be $$ \begin{cases} c\lambda_1=\lambda_3a\\ b\lambda_1=\lambda_2a\\ \lambda_1+\lambda_2+\lambda_3=1\\ \end{cases} $$ where it is assumed that $a,b,c,\lambda_1,\lambda_2,\lambda_3$ are strictly positive and $a+b+c=1$.
Question: I want to show that $\lambda_1=a, \lambda_2=b, \lambda_3=c$ is the unique solution of the system. I tried various substitution routes but couldn't come up with any clean steps. Could you help?