The treatment of a certain viral disease requires a combination dose of drugs D1 and D2. Each unit of D1 contains 1 milligram of factor X and 2 milligrams of factor Y, and each unit of D2 contains 2 milligrams of factor X and 3 milligrams of factor Y. If the most effective treatment requires 13 milligrams of factor X and 22 milligrams of factor Y, how many units of D1 and D2 should be administered to the patient?
To determine the number of units of D1 and D2 that should be administered to the patient, we can set up a system of equations based on the amount of factor X and factor Y required.
Let's say the number of units of D1 to be administered is represented by variable 'x', and the number of units of D2 is represented by variable 'y'.
Based on the information given, the equation for factor X can be written as:
1x + 2y = 13 (equation 1)
Similarly, the equation for factor Y can be written as:
2x + 3y = 22 (equation 2)
Now, we can solve this system of equations to find the values of 'x' and 'y'.
Multiplying equation 1 by 2 and equation 2 by 1, we get:
2x + 4y = 26 (equation 3)
2x + 3y = 22 (equation 2)
Subtracting equation 2 from equation 3, we eliminate the 'x' term:
2x + 4y - (2x + 3y) = 26 - 22
y = 4
Now, substituting the value of 'y' into equation 1, we can solve for 'x':
1x + 2(4) = 13
1x + 8 = 13
1x = 5
x = 5
Therefore, the most effective treatment requires administering 5 units of D1 and 4 units of D2 to the patient.
To determine the number of units of drugs D1 and D2 required for treatment, we need to find a combination that provides the necessary amount of factor X and factor Y.
Let's assume that we administer x units of drug D1 and y units of drug D2.
From the information given, each unit of drug D1 contains 1 milligram of factor X and 2 milligrams of factor Y. Therefore, x units of drug D1 would provide x milligrams of factor X and 2x milligrams of factor Y.
Similarly, each unit of drug D2 contains 2 milligrams of factor X and 3 milligrams of factor Y. So, y units of drug D2 would provide 2y milligrams of factor X and 3y milligrams of factor Y.
According to the problem, we need 13 milligrams of factor X and 22 milligrams of factor Y for the most effective treatment.
Equating the amount of factor X, we have:
x + 2y = 13 (Equation 1)
Equating the amount of factor Y, we have:
2x + 3y = 22 (Equation 2)
We now have a system of equations (Equation 1 and Equation 2) to solve for x and y.
To solve this system of equations, there are multiple methods, including substitution, elimination, or matrix methods. We'll use the elimination method here:
We'll multiply Equation 1 by 2 and Equation 2 by -1 to eliminate the x term:
2*(x + 2y) = 2*13 --> 2x + 4y = 26
-1*(2x + 3y) = -1*22 --> -2x - 3y = -22
Adding these two equations cancels out the x term:
(2x + 4y) + (-2x - 3y) = 26 + (-22)
Simplifying:
y = 4
Now that we have the value of y, we can substitute it back into Equation 1 to find x:
x + 2(4) = 13
Simplifying:
x + 8 = 13
x = 5
Therefore, to treat the viral disease, 5 units of drug D1 and 4 units of drug D2 should be administered to the patient.