On two investments totaling $10,500, brian lost 6% on one and earned 8% on the other. if his net annual receipts were $497, how much was each investment?

Let
x = the amount invested on the winner, then
10500-x = amount invested on the losing investment.

We know that the net profit is $497, so
0.08x - 0.06(10500-x) = 497
Solve for x:
0.14x = 497+630=1127
x=1127/0.14=8050

Therefore Brian invested $8050 on the investment that earned 8% and $2450 on the one that lost 6%.