Answer:
- From Commercial spray X: 4 liters from Chemical A, 8 liters from Chemical B and 8 liters from Chemical C.
- From Commercial spray Y: 18 liters of Chemical C
- From Commercial spray Z: 8 liters of Chemical A and 8 liters of Chemical B.
Step-by-step explanation:
Since we need a lot of chemical C, we are going to leave Commercial spray Y for last in case we lack in case we need more Chemical C, which is supplied by spray Y.
We need to cover the demand of both chemical A and B. For each liter of Chemical A we get 2 liters of Chemical B by using Commercial spray X. And for each liter of Chemical A we get 2 liters of Chemical B by using spray Z.
We will call X the number of units of Commercial spray X used, where each unit contains 1 liter of Chemical A, 2 liters of Chemical B and 2 iters of Chemical C. Y is the number of units of Commercial spray Y used, containing each of them 1 liter of Chemical C, and Z the number of units of commercial spray Z used, on units containing both 1 liter of Chemical A and Chemical B.
We can represent the amount of liters we have for each chemical on a vector of the form <em>(a,b,c)</em> where a reresents the amount of Chemical A, b the amount of chemical B, and c the amount of chemical C. We want<em> (a,b,c)</em> to be equal to (12,16,26), in other words, we want a to be 12, b to be 16 and c to b 26. Furthermore we can obtain <em>(a,b,c) </em>by using this equation
(a,b,c) = X * (1,2,2) + Y * (0,0,1) + Z * (1,1,0) = (1*X+0*Y+1*Z,2*X+0*Y+1*Z,2*X+1*Y+0*Z) = (X+Z,2*X+Z,2*X+Y)
Thus, we have
- a = X+Z = 12
- b = 2*X+Z = 16
- c = 2*X+Y = 26
We can substract the second equation with the first one to obtain the value of X:
X = b-a = (2*X+Z)-(X+Z) = 16-12 = 4
Replacing X by 4, we obtain on the first expression that 4+Z = 12, hence, Z = 8. Since X is 4, 2*X+Y = 8+Y = 26. This gives us that Y must be 18.
We conclude that we need the following amount of Chemical A, B and C:
- From Commercial spray X: 4, 4*2 = 8 and 4*2 = 8 liters respectively
- From Commercial spray Y: 18 liters of Chemical C
- From Commercial spray Z: 8 liters of Chemical A and 8 liters of Chemical B.
I hope i could help you!
