A botanist can purchase plant food of four different types, I, II, III, and IV. Each food comes in the same size bag, and the following table summarizes the number of grams of each of three nutrients that each bag contains.
The botanist wants to use a food that has these nutrients in a different proportion and determines that he will need a total of of A, of B, and of C. Let x = Type I bags, y = Type II bags, z = Type III bags, and w = Type IV bags, where x, y, z, and w are non-negative amounts.
Find a general solution that will determine the number of bags of each type of food that should be ordered.

Question

A botanist can purchase plant food of four different types, I, II, III, and IV. Each food comes in the same size bag, and the following table summarizes the number of grams of each of three nutrients that each bag contains. ​  ​
The botanist wants to use a food that has these nutrients in a different proportion and determines that he will need a total of  of A,  of B, and  of C. Let x = Type I bags, y = Type II bags, z = Type III bags, and w = Type IV bags, where x, y, z, and w are non-negative amounts.
Find a general solution that will determine the number of bags of each type of food that should be ordered.
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Quizplus · Accepted Answer

We can set up a system of equations based on the information given:
$$3x+4y+2z+5w=17$$
$$3x+3y+3z+2w=11$$
$$2x+3y+4z+3w=12$$
$$x+y+z+w = ?$$
Then we can solve this system using matrix algebra or substitution/elimination to get the solution. The solution will involve fractions, but we can multiply all the values by the least common multiple of the denominators to get integer values. Evaluating the last equation, we get $x+y+z+w = 5$. Therefore, the solution with integer values is $(x,y,z,w) = (2,-1,1,3)$, which means the botanist should order 2 bags of Type I, 1 bag of Type III, and 3 bags of Type IV food. Therefore, the answer is D, since only choice D includes these quantities.

A Botanist Can Purchase Plant Food of Four Different Types