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System of Equations

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?
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1 Answer

Joey:
 
One unit of D1 contains 1 mg of X and 2 mg of Y.  One unit of D2 contains 2 mg of X and 3 mg of Y
 
Suppose you choose a units of D1 and b units of D2.  Then you get a mg of X and 2a mg of Y (from D1) and 2b mg of X and 3b mg of Y from D2.  So you get a total of a + 2b  mg of X and 2a + 2b mg of Y.  What you want is
 
  a + 2b = 13
 
2a + 3b  = 22
 
Multiply first equation by 2 throughout:  2a + 4b = 26.  Subtract the second equation to get b = 4.  That gives a + 8 = 13, so that a = 5.
 
Answer: Take 5 units of D1 and 4 units of D2.  Check for yourself that you get 13 mg of X and 22 mg of Y, total, by doing so.  (I have!)
 
Dattaprabhakar (Dr. G.)

Comments

Thank you for your answer! It makes total sense now. However, I am still a bit perplexed on why you put 13 and 22 in their respective places. Since: Effective treatment =13mgX + 22mgY, doesn't X and Y have to be a combination from both D1 and D2? The math makes perfect sense, but getting from the words to the equation still puzzles me.
 
Thanks,
Joey
Joey:
 
See that "the most effective treatment requires 13 milligrams of factor X and 22 milligrams of factor Y". You can not get these directly but in certain preassigned combinations, though D1 and D2. The objective therefore is to get the correct units of D1 and D2 to provide the right amounts of X and Y factors.  IT IS A COMBINATION (obtained) FROM BOTH D1 AND D2.
 
Dattaprbhakar (Dr. G.)
 
P.S. Actually, your "13 mg X + 14 mg Y" does not make sense as an algebraic sum, though I understand what you mean.  It is like  combining "apples and oranges".