An oil company operates two Southern refineries. The Alabama facility produces 500 barrels of high-grade oil and 200 barrels of medium-grade oil per day. The Texas facility produces 600 barrels of high-grade and 400 barrels of medium-grade oil per day. By operating each facility for a certain number of days, the company produces exactly 15,000 barrels of high-grade and 8400 barrels of medium-grade oil. How many days must each facility operate to produce the desired result?

To solve this problem, we can set up a system of equations based on the given information.

Let's call the number of days the Alabama facility operates as x, and the number of days the Texas facility operates as y.

Based on the information provided, the total amount of high-grade oil produced can be calculated as:
500x + 600y = 15,000 (equation 1)

Similarly, the total amount of medium-grade oil produced can be calculated as:
200x + 400y = 8,400 (equation 2)

We now have a system of two equations with two variables. To solve these equations, we can use a method called substitution or elimination.

Option 1: Substitution Method:
1. Solve equation 1 for x:
x = (15,000 - 600y) / 500

2. Substitute the value of x in equation 2:
200((15,000 - 600y) / 500) + 400y = 8,400

3. Simplify and solve for y:
(30,000 - 1,200y) / 5 + 400y = 8,400
(30,000 - 1,200y + 2,000y) / 5 = 8,400
30,000 + 800y = 42,000
800y = 12,000
y = 15

4. Substitute the value of y obtained in step 3 into equation 1 and solve for x:
500x + 600(15) = 15,000
500x + 9000 = 15,000
500x = 6000
x = 12

Therefore, the Alabama facility must operate for 12 days, while the Texas facility must operate for 15 days to produce the desired result.

Option 2: Elimination Method:
1. Multiply equation 2 by 3 to make both equations have the same coefficient for y:
600x + 1200y = 25,200 (new equation 2)

2. Subtract equation 2 from equation 1 to eliminate y:
500x + 600y - (600x + 1200y) = 15,000 - 25,200
-100x + 600y = -10,200
-100x = -600y - 10,200

3. Divide both sides of the equation by -100 to solve for x:
x = (600y + 10,200) / 100
x = 6y + 102 (equation 3)

4. Substitute the value of x from equation 3 into equation 1:
500(6y + 102) + 600y = 15,000
3000y + 51,000 + 600y = 15,000
3600y + 51,000 = 15,000
3600y = -36,000
y = 10

5. Substitute the value of y obtained in step 4 into equation 3 and solve for x:
x = 6(10) + 102
x = 162

Therefore, the Alabama facility must operate for 162 days, while the Texas facility must operate for 10 days to produce the desired result.

Note: In option 2, the negative value for x is due to the subtraction while eliminating y. This does not affect the validity or interpretation of the solution.

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