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To solve this I am going to create three equations for what each person bought.
a will represent apples, b will represent berries, and c will represent cherries.

34 = 2a + 1b + 4c
35 = 3a + 2b + 2c
49 = 5a + 3b + 2c

Solve for on variable in the first equation.

b = 34 - 2a - 4c

Plug that into the second equation and solve for a different variable.

35 = 3a + 2(34 - 2a - 4c) + 2c
35 = 3a + 68 - 4a - 8c + 2c
35 = -a + 68 - 6c
-33 = -a - 6c
33 = a +6c
a = 33 - 6c

It gets a little confusing here but you plug in what both a and b are equal to until you have solved for c.

49 = 5a + 3b + 2c
49 = 5(33 - 6c) + 3b + 2c   substitute a with 33 - 6c
49 = 165 - 30c + 3b +2c
49 = 165 - 28c + 3b
-116 = -28c + 3(34 - 2a - 4c)   substitute b with 34 - 2a - 4c
-116 = -28c + 102 - 6a - 12c
-116 = -40c + 102 - 6(33 - 6c)   again substitute a with 33 - 6c
-116 = -40c + 102 - 198 + 36c
-116 = -4c - 96
-20 = -4c
4c = 20
c = 5

a = 33 - 6c
a = 33 - 6(5)
a = 33 - 30
a = 3

b = 34 - 2a - 4c
b = 34 - 2(3) - 4(5)
b = 34 - 6 - 20
b = 8

Apples cost $3 per pound
Berries cost $8 per pound
Cherries cost $5 per pound