Arancher constructs two rectangular horse pastures that share a side, as shown. the pastures are enclosed by 1050 feet of fencing. each pasture has an area of 15,000 square feet.

The complete question in the attached figure

we know that
area each pasture=15000 ft²
area each pasture=x*y
so
15000=x*y-----> equation 1

perimeter two rectangular pastures=1050 ft
perimeter two rectangular pastures=2*[2x+y]+y----> 4x+2y+y---> 4x+3y
so
1050=4x+3y----> divide by 3----> 1050/3=(4/3)x+y
clear variable y
y=350-(4/3)x-----> equation 2

the answer part a) 
y=350-(4/3)x------> it was proved

Part b)Find the possible lengths and widths of each pasture.
substitute equation 2 in equation 1
15000=x*y-----> 15000=x*[350-(4/3)x]
15000=350x-(4/3)x²----> multiply by 3----> 45000=1050x-4x²
4x²-1050x+45000=0

using a graph tool----> to resolve the second order equation
see the attached figure

the solution is 
x=53.942 ft
x=208.558 ft

15000=x*y-----> y=15000/x

for x=53.94 ft
y=278.09 ft

for x=208.56 ft
y=71.92 ft

the possible lengths and widths of each pasture are
case 1
x=53.94 ft
y=278.09 ft

case 2
x=208.56 ft
y=71.92 ft