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Let $r$ and $b$ be the number of red and blue marbles originally in the bag respectively. After $1$ red marble is removed, there are $r+b-1$ marbles left in the bag and $r-1$ red marbles left. So $$\frac{r-1}{r+b-1}=\frac{1}{7}.$$ When $2$ blue marbles are removed, there are $r$ red marbles and $r+b-2$ total marbles left in the bag. So $$\frac{r}{r+b-2}=\frac{1}{5}.$$ Cross multiplying for each yields $$\begin{array}{rl}7r-7=r+b-1& ⟹7r-6=r+b\\ 5r=r+b-2& ⟹5r+2=r+b.\end{array}$$ We can equate each of these expressions to yields $$7r-6=5r+2⟹2r=8⟹r=4⟹b=18.$$ Therefore, the total number of marbles is $$r+b=4+18=\boxed{22}$$