systems of equations - Knowing that a feasible solution exists and has a finite optimal solution

I have the following linear programming problem:constraints:$x_1,x_2,x_3\geq200$$0.45x_1+0.41x_2+0.5x_3 \leq 960$$x_1+x_2+x_3 \leq 2000$$ x_2+x_3 \leq x_1$objective functions:max $0.35 x_{1}+0.41 x_{2} + 0.37 x_3$min $0.45x1+0.41x_2+0.5x_3$How can I tell without solving the problem numerically that there is a feasible solution for both objective functions and a finite optimal solution?Any advice about the theorem or the intuition would be greatly helpful!...Read more

linear programming - Finding numerical values to an equation describing a hyperplane or a plane (any software suggestion?)

The following equation$$0.27a+0.1b+0.13c=70$$can admit many solution. Is there any software/methods I can use so that I can have a large list of all the possible numerical solutions to this equation? The background to such a problemSo I am designing for my friend a recipe, and I know chicken, yoghurt and eggs, contain 0.27g, 0.1g and 0.13g of proteins per gram respectively. Suppose I would like to have 70g of proteins, what are the combinations of the respective amount of food to make up the desired amount of proteins? In principle I can just ...Read more

systems of equations - Multiple inputs, multiple outputs; solving when you have simple linear models

I am working on a mathematical problem related to a steady state controls problem, and I think this might be the place to ask this. I've figured out some of the simple cases, and am wondering where might the best place to look for a more generic solution, since the real problem involves more inputs (over 50; this is a problem for multiple light sources and sensors, but could be applied to multi-zone heating ovens, for example).The problem is simplified, since the outputs are linearly dependent on the inputs.For a single input, and three output...Read more

systems of equations - Solve $x^{3} = 6+ 3xy - 3 ( \sqrt{2}+2 )^{{1}/{3}} , y^{3} = 9 + 3xy(\sqrt{2}+2)^{{1}/{3}} - 3(\sqrt{2}+2)^{{2}/{3}}$

Solve the system of equations for $x,y \in \mathbb{R}$ $x^{3} = 6+ 3xy - 3\left ( \sqrt{2}+2 \right )^{\frac{1}{3}} $ $ y^{3} = 9 + 3xy(\sqrt{2}+2)^{\frac{1}{3}} - 3(\sqrt{2}+2)^{\frac{2}{3}}$I just rearranged between those equations and get $ \frac{y^{3}-9}{x^{3} -6} = (\sqrt{2}+2)^{\frac{1}{3}}$ then I don't know how to deal with it.Please give me a hint or relevant theorem to solve the equation.Thank you, and I appreciate any help. Furthermore I get an idea how about we subtract two equation and get $y^{3}-x^{3} = 3 + 3xy((\...Read more

chinese remainder theorem - CRT - non-linear system of equations

I don't know how to solve system of equations using CRT when there is some quadratic/cubic variable. For example: System 1: $$\boxed{x^3 \equiv 1 \pmod{3}}$$ $$12x \equiv 9 \pmod{15}$$ System 2: $$3x \equiv 6 \pmod{9}$$ $$\boxed{x^2 \equiv 1 \pmod{4}}$$ $$4x \equiv 2 \pmod{5}$$I think (please correct me if I'm wrong) the quadratic equation from System 2 can be rewritten into linear equations as $x \equiv \pm 1 \pmod{2}$.How can I rewrite the cubic equation from System 1 into linear equations to be able to use CRT?...Read more

quadratics - Possibility of solving a certain 4D system with 4 equations with certain methods

Can the system \begin{align}x+y& =a\\ x^2+b& =z^2\\y^2+c&=w^2\\dzy&=wx\end{align} (with $x,y,z,w,a,b,c,d >0$) be solved for $x, y, w$ and $z$ by hand without cubics (which I could solve... but yikes) or quartics, and using only elementary functions? I was challenged to find a simple solution but after alot of manipulating I'm starting to think it's not possible. I know that once you solve for one variable you can use one of the first three equations to isolate another easily with only roots and etc., and then do that again wi...Read more

systems of equations - Solve $ab + cd = -1; ac + bd = -1; ad + bc = -1$ over the integers

I am trying to solve this problem: Solve the system of equations \begin{align}\begin{cases}ab + cd = -1 \\ac + bd = -1 \\ad + bc = -1\end{cases}\end{align} for the integers $a$, $b$, $c$ and $d$.I have found that the first equation gives $d = \dfrac{-1-cd}{a}$, which gives $a\neq0$. Other than that, I don't know where to start.Tips, help or solution is very appreciated. Thanks!...Read more

rational functions - Elimination of variables from a system of equation.

Suppose $a,b,c,d,e,f,g,h,i$ are real numbers. Consider the system of equation\begin{equation}\begin{pmatrix}1 \\ 1 \\ 1\end{pmatrix}= \begin{pmatrix}a + bx + cy \\ \frac{d}{x} + e + fz \\ \frac{g}{y} + \frac{h}{z} + i\end{pmatrix}.\end{equation}We need to eliminate the variables $x,y$ and $z$ from the above (assuming that solution for the variables $x,y$ and $z$ exists) and get a single equation in $a,b,c,d,e,f,g,h,i$. I do not know it is possible or not. I hope this is possible but I am unable to get the required. Any suggestion or further h...Read more

systems of equations - Need help solving a simultaneous quadratic

I'm having a hard time with simultaneous quadratics. I've got this question and don't know how to start off on solving it. I know how to solve linear simultaneous equations and that's about it.$$x^2 + y^2 = 25\\y - 2x = 5$$Ok, so I've managed to square the second equation and got$$ (y-2x)(y-2x) = 25 = y^2 -4xy + 4x^2$$I also find it hard to recognise when a pair of equations is a quadratic. How do I identify when it is a quadratic simultaneous equation?...Read more