Math: Systems Of Equations help
Summary: How can I find the closest possible positive values to solve a system of equations for many variables and equations?
First let me give a concrete example of my problem:
A concrete example:
If I have a given set of ingredients, let's say apples, chicken, and butter
Apple:______fat: 0g_____protein: 1g _____carbs: 16g
Chicken:___ fat: 4g_____protein: 20g____carbs: 0g
Butter: ____ fat: 100g __protein: 0_______carbs: 0g
And I want to eat a given amount of nutrients, let's say 20g of fat, 22g of protein, and 10g of carbs .
Fat 20 ≈ 0a + 4c + 100b
Protein 22 ≈ 1a + 20c + 0b
Carbs 10 ≈ 16a + 0c + 0b
a= Amount of apple (1=100grams of apple)
c= Amount of chicken
b= Amount of butter
How much of each food should I eat to get as close as possible to my desired nutrients?
I need to be able to solve problems like these where the amount of ingredients (variables) can vary, and the amount of nutrients(equations) can vary. Though for now, it is less important that the amount of equations vary as long as I can solve for three equations (fat, carbs, protein).
So here are my abstract questions:
1. How do I solve systems of equations where there can be many equations and many variables.
Ex: We should be able to solve for any of these:
6 ≈ 2x+1y +12z
40≈1x+7y +6z +8g + 10a +7f
60 ≈ 2x+1y +12z +9g + 3a + 8f
14≈5x+2y +3z +4g +7a +3f
6 ≈ 2x+1y +12z
68 ≈ 2x+1y +12z
2. The variables can't equal negative values since you can't have a negative amount of food. If the only solution uses a negative value, how can I find an option with the closest possible positive amount?
This is what I mean by closest possible:
For example's sake, pretend that the only true solution to this system of equations gives us at least one negative variable:
6 ≈ 2x+1y
If x=3 and y =1 , then the difference between our left side and and right side is an absolute value of 3.
6 ≈ 7 __________+1
Total difference: 3
But are there values for x and y that would give a smaller total difference? How do we find that value?