Mr R.

# Matrices 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).

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.