Phyllis invested $10,500 a portion earning a simple interest rate of 9(1/5)% per year and the rest earning a rate of 9% per year, after one year the total interest earned on these investments was $957.00. how much money did she invest at each rate?

Total investment  = $10,500

Let x = amount of first investment, and y the amount of the second investment.

First investment:
Interest rate = 9 (1/5)% = 0.092
Earned interest = 0.092x

Second investment:
Interest rate = 9% = 0.09
Earned interest = 0.09y

Total interest after one year is $957.00, therefore
0.092x + 0.09y = 957       
 or
1.0222x + y = 10633.33   (1)

Also,
x + y = 10500               (2)

Subtract (2) from (1).
0.0222x = 133.33
x = 6000
y = 10500 - x = 4500

The first investment is $6,000 at 9 (1/5)% rate;
The second investment is $4,500 at 9% rate.