A small bag of trail mix contains 3 cups of dried fruit and 4 cups of almonds.A large bag contains 41/2 cups of dried fruit and 6 cups of almonds. Write and solve a system of linear equations to find the price of 1 cup of dried fruit and 1 cup of almonds if the Price of small bag is $6 while $9 for large bag.        

Let's call:
f = price of 1 cup of dried fruit
a = price of 1 cup of almonds

In order to build the linear system, you need to consider that the total price of a bag is given by the sum of the price of cups times the number of cups in each bag, therefore:

[tex] \left \{ {{3f + 4a = 6} \atop {41/2 f + 6a = 9 }} \right. [/tex]

Solve for a in first equation:
a = (6 - 3f) / 4

Then substitute in the second equation:
41/2 f + 6 · (6 - 3f) / 4 = 9
41/2 f + 9 - 9/2 f = 9
16 f = 0
f = 0

Now, substitute this value in the formula found for a:
a = (6 - 3·0) / 4
   = 3/2 = 1.5

Hence, the cups of dried fruit are free and 1 cup of almond costs 1.5$