Having trouble getting started on a word problem involving dual APR's

the question is: "Cathy just won $10,000 in the Oregon state lottery. She wants to put some of the money in an account where it is accessible and to commit the rest to a long-term investment. She has been considering a savings account that has been paying an annual percentage rate (APR) of 2.64% and a mutual fund that has been paying an APR of 8.9%. At the end of one year, Cathy hopes to earn $750. How much money must she invest in each type of account?"

i have manually figured out that $7763.51 must be invested in the 8.9% mutual fund, and $2236.49 must be invested in the 2.64% saving account to yield a profit of $750 on $10,000. The problem is I sort of just ballpark guessed one variable to i narrowed down, and then fine tuned. What would be a systematic and intelligent approach to this problem? thank you