I have a system of linear equations that I need a solution for that is strictly positive. I have 4 solutions and 4 unknowns, and the solution I obtain for my current system involves negative numbers. For illustrative purposes, here is the system: $$x_1+ x_2 + x_3 + x_4 = 1$$ $$0.2895 x_1+1.2312 x_2 + 0.8360 x_3+ 0.5515 x_4 = 0.665$$ $$0.2942 x_1+ 0.5828 x_2 + 0.4561 x_3 + 0.4006 x_4 =0.47$$ $$7,969,147 x_1 +21,685 x_2+ 2,656,382 x_3 +7,083,686 x_4 = 1,328,191.2$$
I can vary the coefficients in my equations a little bit and I can vary the LHS of two of my equations (a little bit).
I had originally tried to treat this problem as a Linear Programming problem but was unable to find solutions for my given coefficients even while varying the LHS of the equations.
So my question is: How can I go about selecting coefficients from within a certain range without feeling like I am just stabbing in the dark? Is there a name for this type of problem? It would be a Linear Programming problem but I want to be able to vary the coefficients in my constraints until I find a solution and I also don't have an objective function.
Below is the current system I'm considering. All of my coefficients must be positive. I can make adjustments to the coefficients on line 2 by less than 0.5 each. I can change coefficients in line 4 drastically (by millions). I can make no adjustments to coefficients in lines 1 or 3. $$x_1+ x_2 + x_3 + x_4 = 1$$ $$ 0.64 \geq 0.2895 x_1+1.2312 x_2 + 0.8360 x_3+ 0.5515 x_4 \leq 0.67$$ $$0.44 \geq 0.2942 x_1+ 0.5828 x_2 + 0.4561 x_3 + 0.4006 x_4 \leq 0.48$$ $$7,969,147 x_1 +21,685 x_2+ 2,656,382 x_3 +7,083,686 x_4 = 1,328,191.2$$