Matrices question

Question

Hi I'm am struggling to answer the last question in my textbooks chapter...

Three crop sprays are manufactured by combining chemicals A, B and C as follows:

One barrel of spray P contains 1 unit of A, 3 units of B and 4 units of C.

One barrel of spray Q contains 3 units each of A, B and C.

One barrel of spray R contains 2 units of A and 5 units of B.

To control a certain crop disease, a farmer requires 6 units of chemical A, 10 units of chemical B and 6 units of chemical C. How much of each type of spray should the farmer use?

I did find a similar question already answered here however I couldn't get the right answer using that working.

I know some sites like to see the user attempt the question first... skip the rest if you don't care.

What I have tried is:

1. creating a 3x3 matrix with the amount of each chemical in each barrel
2. multiplying this matrix by a column matrix with entries A, B, and C (this is done because multiplying the 2 together gives the set of simultaneous equations).
3. Setting this equal to the column matrix 6,10,6 (the required amount of each chemical).

To solve I multiplied both sides by the inverse of the 3x3 matrix and solved (right side becomes the identity and left side should be the answer).

1. The answer I get:
2. A=92/39
3. B=10/39
4. C=28/39

Obviously my answer should be in terms of P, Q and R but I don't know how to set up the equation to provide this.

Answer 1

If I understand the problem correctly, we need

xP + yQ +zR = 6A+10B+6C

Ax + 3Bx +4Cx +3Ay + 3By + 3Cy +2Az+ 5Bz = 6A+10B+6C

A(x+3y+2z) = 6A

B(3x+3y+5z)=10B

C(4x+3y) = 6C

[1 3 2 6]

[3 3 4 10]

[4 3 0 6]

Finding A^-1[6 10 6]^t ==> x = 8/13, y = 46/39 and z = 12/13

Need 8/13 of a barrel of P, 46/39 barrels of Q and 12/13 of a barrel of R