The owner of a candy store wants to mix some peanuts worth $3 per pound, some cashews worth $10 per pound, and some brazil nuts worth $9 per pound to get 50 pounds of a mixture that will sell for $6.80 per pound. she uses 10 fewer pounds of cashews than peanuts. how many pounds of each did she use?

Let
x = pounds of peanuts
y = pounds of cashews
z = pounds of Brazil nuts.

The total pounds is 50, therefore
x + y + z = 50                   (1)

The total cost is $6.60 per pound for 50 pounds of mixture.
The total is equal to the sum of the costs of the different nuts.
Because the cost for peanuts, cashews, and Brazil nuts are $3, $10, and $9 respectively, therefore
3x + 10y + 9z = 50*6.8
3x + 10y + 9z = 340         (2)

There are 10 fewer pounds of cashews than peanuts, therefore
x = y + 10                         (3)

Substitute (3) into (1) and (2).
y + 10 + y + z = 50
2y + z = 40                     (4)
3(y + 10) + 10y + 9z = 340
13y + 9z = 310                (5)

From (4),
z = 40 - 2y          (6)
Substitute (6) into (5).
13y + 9(40 - 2y) = 310
-5y = -50
y = 10
z = 40 - 2y = 40 - 20 = 20
x = y + 10 = 20

Peanuts:     20 pounds
Cashews:   10 pounds
Brazil nuts: 20 pounds

How many groups are there?