When I present partial fraction decomposition, there are typically two approaches to solve for the coefficients of the fractions:
Clear the fractions, multiply out the products, and equate coefficients (of the indeterminate). This leads to a system of linear equations.
Clear the fractions, and substitute several values for the indeterminate. This leads to a different system of linear equations.
Is there a good reason to prefer one method over the another? Is there another situation (perhaps in differential equations) that I'm not thinking of where one method is superior to the other? Is there a good pedagogical reason to present one method over another?
Considering the theory of quotient fields and polynomials, both methods should work (provided the partial fractions have been computed correctly so that a solution exists).
I may consider also asking this question on Mathematics Educators, but I thought to start here.