You are given 10,000 to invest in either 5% tax free bonds or 10% taxable bonds. your total investment income for a year was $675. how much was invested at each rate?

At 5%... $3500
at 10%... $6500

Let us suppose, I invested $x in 5% tax free bonds 
then $(10,000-x) is invested in 10% tax free bonds 
Invested income in the 5% tax free bond is 5x/100 
and invested income in the 10% tax free bond is 10(10,000-x)/100 
Total investment income for a year was $675 
So we can say, 
10(10,000-x)/100 + 5x/100 =675 
or, 100000 - 10x + 5x =67500 
or, 5x = 100000 - 67500changing sides or, x = 32500/5 
 = 6500 
so, 10000-x = 10000 - 6500 = 3500 
Henceforth, the amount of investment at 5% rate is $6500 and at 10% rate is  $3500 .