how many acres of each crop should he plant?

Question

A farmer has 1310 acres of land on what you guys corn wheat in soybeans. It cost $45 per acre to grow corn 60 to grow wheat and 50 to go soybeans because of the market demand. The farmer will grow twice as many acres of wheat as of corn. He has allocated $70,000 for the cost of growing his crop how many acres of each crop should he plant?

Answer 1

45c+60w+50s=70000

9c+12w+10s=14000

c+w+s=1310

w=2c

9c +12w+10s = 14000

9c +9w +9s = 11790

3w +s = 2210

s =2210-3w

c+w+2210-3w = 1310

c -2w =1310-2210=1900

c = 2w-900

c=4c-900

3c=900

c=300 acres for corn

w=2c= 600 acres for wheat

s=1310-c-w=410 acres for soybeans

check the solution:

9c+12w+10s =14000

9(300)+12(600)+10(410)

=2700+7200+4100

=14000

this seems to be the answer, as the constraints limit

the answer to only one solution

Yet the problem as designed seems to be a linear programming problem

with an objective function and constraints that would normally give

points to be evaluated, such as all exterior points (c,w,s) on the boundary of

a polygon, like (0,0,0), (0,1310,0),(0,0,1310),(1310,0,0), etc. but the constraints seem to already lead to just one possible solution, as if overdetermined for linear programming