Mixing sherry from two casks

Cask $A$ contains a mixture of two kinds of sherry in the ratio $2:7$, and cask $B$ contains the same two kinds in the ratio $1:5$. How many gallons must be taken from each cask so that the combined mixture consists of $2$ gallons of one kind and $9$ gallons of the other?

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. II. Proportion, Examples II, No. 20