HOW TO SOLVE LINEAR EQUQTIONS WITH FIVE UNKNOWN
It's clear that the percentages exceed 100%, so I conclude that the dry matter content includes the crude ingredients, which are supplemented by another ingredient that is not specified. I also conclude that the dry matter content indicates that water is present, because none of the percentages for dry matter are 100%.
First, remove water content and recalculate percentages: to do this we divide each percentage of the components by the percentage of dry matter. The result is expressed as a percentage. This enables us to mix the feed on a dry basis.
The adjusted percentages are:
CORN: CP: 9.4909%, CF: 2.3812%, CFB: 1.1339%
RICE: CP: 13.4427%, CF: 13.7776%, CFB: 8.3150%
SBM: CP: 48.0682%, CF: 6.3636%, CFB: 5.1136%
WHEAT: CP: 18.5698%, CF: 5.1903%, CFB: 8.5352%
COPRA: CP: 20.7792%, CF: 13.2035%, CFB: 12.4459%
We also have to adjust the percentages on the required feed:
CP: 18.1818%, CF: 4.5455%, CFB: 13.6364%
The last letter of each ingredient will be used to signify the fraction of that ingredient needed to make up the required feed (N, E, M, T, A). So we can write:
CP REQUIREMENT: 9.4909N+13.4427E+48.0682M+18.5698T+20.7792A=18.1818
CF REQUIREMENT: 2.3812N+13.7776E+6.3636M+5.1903T+13.2035A=4.5455
CFB REQUIREMENT: 1.1339N+8.3150E+5.1136M+8.5352T+12.4459A=13.6364
N+E+M+T+A=1 (The ingredient fractions must add up to 1.)
[To show these equations are valid, let's move away from percentages and consider actual amounts in the dry mix. For example, n grams of corn contains 9.4909n/100 grams (0.094909n grams) of crude protein. Similarly for the other ingredients: e grams of rice contains 13.4427e/100 grams (0.134427e grams) of crude protein. When we add together the crude protein contributions for all the ingredients we get the required amount, 18.1818x/100 (0.1818x grams), where x grams is the weight of the dry mix=n+e+m+t+a. So we can multiply through by 100 to leave the percentage numbers; if we divide through by x we get n/x, e/x, etc., and we can replace these by N, E, etc., where these are fractions of the total feed: N=n/x, E=e/x, ... X=x/x=1.]
There are five variables but only four equations. The amount of copra is A=1-(N+E+M+T), so the equations can be reduced to omit A:
CP REQUIREMENT: 11.2883N+7.3366E-27.2890M+2.2094T=2.5974
CF REQUIREMENT: 10.8222N-0.5741E+6.8398M+8.0132T=8.6580
CFB REQUIREMENT: 11.3120N+4.1308E+7.3323M+3.9107T=-1.1905
This last equation is suspicious because all the values on the left are positive but the value on the right is negative. Therefore at least one assumption in the logic is false. To resolve this difficulty we have to change the requirements to include inequalities. For example, the mix must contain at least a certain amount of an essential ingredient, or no more than a certain amount, as well as exactly a certain amount. We have no guide in the question to make any assumptions. Nevertheless, we'll continue with the solution and see where it leads us.
The plan now is to treat T as an independent variable or constant, and to find each of N, E and M in terms of T. T is therefore a parameter from which the other fractions can be derived. We can eliminate M from CP and CF:
And we can similarly eliminate M from CF and CFB:
We now have two equations involving N, E and T, and we can eliminate E between them and so find N in terms of T:
If we substitute this value of N in 1.9795N-32.4637E+32.0059T=71.6253, we can find E in terms of T:
Substituting for E and N we can find M then A: using the CP requirement equation:
We now have N, E and M in terms of T. We can find A from A=1-(N+E+M+T):
All the ingredients have now been found in terms of T (wheat). These values are based on equality in the ingredient equations.
N=0.8813-0.7148T, E=0.9423T-2.1526, M=0.0386T-0.3093, A=2.5806-1.2661T.
These are all supposed to be positive fractions<1, and clearly they are not! This implies that the exact amounts required in the mix cannot be achieved. I've checked the arithmetic and logic fully and I can find no errors, so it does not appear to be possible to mix the desired quantities of the essential ingredients. It may be possible to get the right quantity of one particular ingredient at the expense of the others, or with an excess of at least one of the other ingredients. An excess could be as undesirable as a deficiency. Read More: ...