I'm not sure how to title this question, so please edit the title if you know better.
I have an equation involving sums of functions: $$ -\sum_{j=i+1}^NQ_jC_{i,j}e^{-Q_jt}=-Q_i\sum_{j=i+1}^NC_{i,j}e^{-Q_jt}+f\sum_{j=i+1}^{N}\sum_{k=i+1}^jC_{k,j}e^{-Q_jt}$$ where each $Q_j$ is unique. I want to solve for the coeffiecients $C_{i,j}$. I know I can produce a solution by considering each exponential function separately i.e. dropping the summation on j: $$ -Q_jC_{i,j}e^{-Q_jt}=-Q_iC_{i,j}e^{-Q_jt}+f\sum_{k=i+1}^jC_{k,j}e^{-Q_jt}$$
I am struggling to articulate the justification for this. Can someone explain why and when I can or can't do this? I'm specifically looking for language to use in my justification. I feel like this is sensible, but just don't know how to express why.