Turkey = 6
Cornucopia = 8
Native American = 7
Mayflower = 11
Pilgrim = 29


let A = Turkey
let B = Cornucopia
let C = Native American
let D = Mayflower
let E = Pilgrim

Then set up five equations:
2A + B + C = 27
C + B + D + A = 32
D + E + 2C = 54
2D + E + A = 57
2E + B + C = 73

Then rearrange the equations with the similar letters underneath:
2A + B + C = 27
A + B + C + D = 32
2C + D + E = 54
A + 2D + E = 57
B + C + 2E = 73

Since there are as many equations as variables (and none of the equations are equivalent), then there must be a solution. You could eliminate one variable (say E) by solving one of the three equations with E in it (I chose the 4th equation):
  E = 57 - A - 2D
Then substitute for E in the other equations.   That would give you four equations with just 4 variables:

2A + B + C = 27
A + B + C + D = 32
2C + D + (57 - A - 2D) = 54
B + C + 2 (57 - A - 2D) = 73

Simplify and rearrange the variables to get:

2A + B + C = 27
A + B + C + D = 32
-A + 2C - D = -3
-2A + B + C - 4D = -41

Now you can eliminate one of the other variables like D in the same manner.
Solve for D in one of the equations and then substitute that expression for D in the other three equations.
You will then have three equations with three variables.

Then continue until you have one equation with one variable and it will be solved.