0
votes
0answers
17 views

Solving a quadratic system of equations for a single variable

I have a quadratic system of $n$ equations that looks like: $$ (A_{j}^{i}y + B_{j}^{i})x_{j}=0 $$ For $i=0...n$. $A_{i,j}$ and $B_{ij}$ are integer matrices and sums over $j$ are implied. $j$ runs ...
0
votes
0answers
20 views

Polynomial systems - conditions for real solution

I was working on the computation of equilibrium points for dynamical systems and arrived in the following system of $n$ polynomials in the variables $(x_1,\ldots,x_n)$ \begin{equation*} ...
5
votes
2answers
89 views

Solve $x+3y=4y^3,y+3z=4z^3 ,z+3x=4x^3$ in reals

Find answers of this system of equations in reals$$ \left\{ \begin{array}{c} x+3y=4y^3 \\ y+3z=4z^3 \\ z+3x=4x^3 \end{array} \right. $$ Things O have done: summing these 3 equations give ...
0
votes
1answer
24 views

Solving for a single variable in a quadratic system

I have a quadratic system of $n$ equations that looks like: $$ A_{ij}x_{j}y + B_{ij}x_{j}=0 $$ For $i=0...n$, where $A_{i,j}$ and $B_{ij}$ are integer matrices and sums over $j$ are implied. Is there ...
3
votes
0answers
38 views

Algorithms for solving overdetermined, homogeneous linear systems with multivariate polynomial coefficients

I would like to solve overdetermined, homogeneous linear systems of equations with multivariate polynomial coefficients, i.e., $Ap=0$ with $A$ an $m\times n$ matrix, $m\gg n$, and $a_{i,j} \in ...
12
votes
2answers
653 views

System of non-linear equations.

I have to find all triplets $(x,y,z)$ that satisfy: $$x^{2012} + y^{2012} + z^{2012} = 3\\x^{2013} + y^{2013} + z^{2013} = 3\\x^{2014} + y^{2014} + z^{2014} = 3$$ I've found the trivial solution ...
0
votes
1answer
60 views

how can I find equation variables?

I have the following equations : $$\begin{cases}K = \frac{B – 3}{20}\\ K = (20S+3)R+S\\ K = 20S^2 + (20N+7)S + N\\ N=S-R \end{cases}$$ - And I have the $B$ values, e.g : 173, 283, 2343, 834343 ...
0
votes
0answers
12 views

unknown function: calculation of coefficients in series expansion up to a given degree

I am trying to solve an functional equation of unknown $h\mapsto h(x)$ ($x\in\mathbb{R}$, in the neighbourhood of $0$): $$\mathcal{F}(h)=\mathcal{G}(h) \qquad (*)$$ (assume $\mathcal{F}$ and ...
3
votes
2answers
42 views

Solving $4y^4 - 4x^4 + x + y = 0$ (equation system of partial derivates)

I need help solving the following equation system: $$ \frac{\partial}{\partial x} = 8xy + 4y^2 + \frac{y}{x^2 + y^2} = 0 $$ $$ \frac{\partial}{\partial y} = 8xy + 4x^2 - \frac{x}{x^2 + y^2} = 0 $$ ...
0
votes
1answer
47 views

How to solve a nonlinear system of three equations involving rational functions?

How do I get $a$, $b$, and $c$? Given $$X=\frac{a+\frac{1}2b}{a+b+c}$$ $$Y=\frac{b(\frac{\sqrt3}{2})}{a+b+c}$$ $$Z=\frac{76a+150b+29c}{255}$$ in other words How do i get $a$, $b$, and $c$ on the ...
3
votes
1answer
27 views

Find the polynomial function

Anybody knows how to find the polynomial function with evaluated values, where if the degree is $n$ I have $n+1$ values of the function like $f(0) = a_0, f(1) = a_1, \ldots, f(n) = a_n$.
2
votes
1answer
106 views

Three phase voltage system of polynomial equations

I'm working with the development of a product in the company where I work. This product measures three phase voltages and currents. I cannot change the circuit because it has been sold for a long time ...
3
votes
2answers
48 views

Prove that system of equation implies statement

How to prove that $$ \begin{cases} x_1 + x_2 + x_3 = 0 \\ x_1x_2 + x_2x_3 + x_3x_1 = p \\ x_1x_2x_3 = -q \\ x_1 = 1/x_2 + 1/x_3 \end{cases} $$ implies $$ q^3 + pq + q = 0 $$ ?
4
votes
2answers
50 views

solving a system of equations dealing with Lorentz transformations

Can anyone help me to find the solutions of this system of equations: $$c^2x^2-v^2y^2=c^2$$ $$y^2-c^2z^2=1$$ $$vy^2+c^2zx=0$$ I know the answer: $$x= \frac{1}{ \sqrt{1- \frac{ v^{2} }{ c^{2} } } } $$ ...
0
votes
0answers
38 views

solving simple system of cubic equations

I know, that there is no general answer as how to solve a system of equations. But mine has a pretty special form. Let $x,a,b \in \mathbb{R}^n$ and $a,b$ are known. I want to find the solution of the ...
4
votes
4answers
109 views

Solve system of nonlinear equations using non-numerical method

Is there any non-numerical method to solve this kind of system of nonlinear equations for $c_1, c_2, x_1, x_2$: $$c_1+c_2 = 1$$ $$c_1x_1+c_2x_2 = 1$$ $$c_1x_1^2+c_2x_2^2 = 2$$ $$c_1x_1^3+c_2x_2^3 = ...
5
votes
2answers
127 views

How find this system $a^2+b^2=3,a^2+c^2+ac=4,b^2+c^2+\sqrt{3}bc=7$

Find the this system real solution $$\begin{cases} a^2+b^2=3\\ a^2+c^2+ac=4\\ b^2+c^2+\sqrt{3}bc=7 \end{cases}$$ I think that one can use Geometry to solve this system. Maybe there exist an ...
1
vote
3answers
115 views

Solving a nonlinear system of equations.

Given that $x,y,z\in\mathbb R$, solve $$\begin{cases}6x^2-12x=y^3\\6y^2-12y=z^3\\6z^2-12z=x^3\end{cases}$$ I've tried adding the equalities but to no avail. I'd add what I've tried, but it'd be ...
5
votes
2answers
246 views

Complex numbers system of equations problem with 5 variables

Let $z_0$,$z_1$,$z_2$,$z_3$ and $z_4$ such that $z_i\in C$ that hold: $$(1)|z_0|=|z_1|=|z_2|=|z_3|=|z_4|=1$$ $$(2)z_0+z_1+z_2+z_3+z_4=0$$ $$(3) z_0z_1+ z_1z_2+z_2z_3+z_3z_4+z_4z_0=0$$ Prove that ...
4
votes
3answers
55 views

Invalid subtraction when solving system of equations?

I'm trying to solve these two equations: $$\begin{cases} 1-4x(x^2+y^2)=0 \\ 1-4y(x^2+y^2)=0 \end{cases}$$ and I tried to do it by subtracting the first equation from the second, yielding ...
5
votes
2answers
55 views

Number of solutions for a system of polynomial equations

Consider the given system of polynomial equations, where all the coefficients are in $\mathbb{C}$: $$\begin{cases} y^n=P(x)\\ Q(x,y)=0\end{cases}$$ I would like to establish that either this system ...
2
votes
2answers
70 views

Does there exist a polynomial function for every n points, whose extremas are these points?

Given $ n $ points in $ \mathbb{R}^2 $, does there exist a polynomial function of any degree, whose extremas include these $ n $ points? Given 3 points: $ P_1 = (0,4), P_2 = (2,2), P_3 = (4,7) $ And ...
0
votes
2answers
47 views

Conditions of the system of equations.

Find m to the equation:$$\left\{ \begin{array}{l}2x^3-\left(y+2\right)x^2+xy=m\,\,(1)\\x^2+x-y=1-2m\,\,(2) \end{array} \right.$$have experience My try: From $(1)$ and $(2)\,\Rightarrow $: ...
0
votes
0answers
22 views

Some solution for a particular system of non-linear equations

I have come across this system of equations where $q>1$ is a real constant, $x_i$ and $y_i$ are real variables and $y_i>0$: $$ \begin{align} 1~~ & = x_0 ~~~~ ~~+ x_1 ~~~~~~~+ x_2 ...
5
votes
2answers
125 views

Why does the following nonlinear system have 21 solutions?

I am curious as to why the following nonlinear system has 21 solutions (according to Wolfram Alpha). $$y+xy^2-x^3+2xz^4=0 \\ -x-y^3-3x^2y+3yz^4=0 \\ -\frac{5}{2}y^2z^3-2x^2z^3-\frac{z^7}{2}=0$$ I ...
0
votes
1answer
25 views

Lowest norm solution to a system of polynomial equations

I have a system of cubic equations: $$0=A_0+A_1 x+A_2 ( x \otimes x ) + A_3( x \otimes x \otimes x )$$ where $\dim A_0 = \dim x$ (so there are as many equations as unknowns). You may assume that the ...
1
vote
0answers
56 views

how to solve these kind of systems $x^2+y^2=z^2; z-y^3=5; xy=z$

Three variable system of equations with three variable with exponents for example $x^2+y^2=z^2$ $z-y^3=5$ $xz=y$
8
votes
5answers
278 views

Given $x+y$ and $x\cdot y$, what is $x^3+ y^3$ ?

I have been looking at an assortment of high school number sense tests and I noticed a reoccurring problem that states what x+y is and what $x\cdot y$ is then asks for $x^3+ y^3$. I want to know how ...
0
votes
1answer
36 views

A question on HCF and Equations. The question is given in the picture below.

Please also mention how you arrived at the answer.
1
vote
2answers
68 views

How find this linear equation $-x_{1}-x_{2}-x_{3}-\cdots+(2^n-1)x_{n}=2^na$?

let $a\in R$,then solve this follow equation: $$\begin{cases} x_{1}-x_{2}-x_{3}-\cdots-x_{n}=2a\\ -x_{1}+3x_{2}-x_{3}-\cdots-x_{n}=4a\\ -x_{1}-x_{2}+7x_{3}-\cdots-x_{n}=8a\\ ...
0
votes
0answers
25 views

Univariate and Multivariate Representation

Lemma Let $\mathbb{E}$ be an extension field and $\mathbb{F}$ the corresponding ground field with q elements. In addition, let $n$ be the dimension of $\mathbb{E}$ over the ground field. Consider the ...
0
votes
2answers
44 views

Solve this system of equations…

Find numbers $a$ , $b$ , $p$ and $q$ so that: $$x^3+15x^2+3x+5=p(x-a)^3+q(x-b)^3$$
4
votes
2answers
191 views

Solving system of multivariable 2nd-degree polynomials

How would you go about solving a problem such as: \begin{matrix} { x }^{ 2 }+3xy-9=0 \quad(1)\\ 2{ y }^{ 2 }-4xy+5=0 \quad(2) \end{matrix} where $(x,y)\in\mathbb{C}^{2}$. More generally, how would ...
1
vote
2answers
350 views

How to solve this symmetric system of equations?

How many solutions are there to this equation? $$\begin{align*} x^2-y^2&=z\\ y^2-z^2&=x\\ z^2-x^2&=y \end{align*}$$
1
vote
1answer
148 views

A nonlinear system of equations

I want to solve the system of equations , $$\left\{ \begin{array}{rcl} y^3 − 9x^2 + 27x − 27 &=& 0\\ z^3 − 9y^2 + 27y − 27 &=& 0\\ x^3 − 9z^2 + 27z − 27 &=& 0\\ \end{array} ...
1
vote
0answers
73 views

How to solve this system equation of polynomials?

I have: $F(x) + G(x) = 1 + F(x)*M,$ $G(x) = T_{1}(x) + T_{2}(x) + ... + T_{N}(x)$ $F(x )x^{a_{i}} = T_{i}(x) \times C_{i}(x) + \sum_{j \leq N} T_{j}(x) \times P_{ji}(x)$ In which $M, N, a_{i}, ...
0
votes
1answer
199 views

Solving a cubic equation system

I got a cubic equation system that contains 3 cubic equations with 3 variables. I want to find the number of solutions and the solutions themselves (as a numerical approximation). Do you know good ...
1
vote
0answers
48 views

Bound for the number of solutions of a polynomial system of equation.

Given integers $d_1,\ldots,d_n$, is there an elementary way to justify that for generic polynomials $f_1,\ldots,f_n\in\mathbb{C}[X_1,\ldots,X_n]$ of degree less or equal than $d_1,\ldots,d_n$, the ...
0
votes
2answers
60 views

How can I show that there are only finitely many solutions for the following system?

$x^2+yz=x$ $y^2+zx=y$ $z^2+xy=z$ I could not do anything to find the solutions. Please give some hints.
3
votes
2answers
124 views

Given ${x^2-x y+y^2 = 15, x y+x+y = 13}$ find the value of $x^2+6y$

Both x and y are real numbers and x > y . Given ${x^2-x y+y^2 = 15, x y+x+y = 13}$ find the value of $x^2+6y$ . I tried solving the second equation to get $y=(13-x)/(x+1)$ and substituted that in ...
7
votes
2answers
216 views

How to solve this system for real $x,y,z$

Find the real values $x,y,z$ such that $$\begin{cases} x+y^2+z^3=21\qquad (1)\\ y+z^2+x^3=71\qquad (2)\\ z+x^2+y^3=45\qquad (3) \end{cases}$$ Thank you everyone. This problem have some nice methods, ...
1
vote
2answers
181 views

Solve a set of non linear Equations on Galois Field

I have the following set of equations: $$M_{1}=\frac{y_1-y_0}{x_1-x_0}$$ $$M_{2}=\frac{y_2-y_0}{x_2-x_0}$$ $M_1, M_2, x_1, y_1, x_2, y_2,$ are known and they are chosen from a $GF(2^m).$ I want to ...
2
votes
5answers
99 views

system of equations solving for positive $a,b,c$

i need help i need to find positive number $a,b,c$ solving this system of equations? $$(1-a)(1-b)(1-c)=abc$$ $$a+b+c=1$$ I found that $0<a,b,c<1$ and I try to solve it by try $(1-a)=a$, ...
5
votes
2answers
216 views

How to solve these three equations?

If α ,β ,γ are three numbers s.t.: $\ α^ \ $ + $\ β \ $ + $ γ \ $ = −2 $\ α^2 \ $ + $\ β^2 \ $ + $ γ^2 \ $ = 6 $\ α^3 \ $ + $\ β^3 \ $ + $ γ^3 \ $ = −5, then $\ α^4 \ $ + $\ β^4 \ $ + $ ...
0
votes
1answer
82 views

Help with non-linear system of equations

This system of equations $$\begin{align} xy+yz+zx & =3 \\ \\ x^4+y^4+z^4 & =3\end{align}$$ How to solve this system of equations? Any help, Plz. Thank all
0
votes
1answer
145 views

Visualise 3 simultaneous cubic equations

I have three equations of the form: $$\frac{i_1^3}{P_1}+i_1(Z_1+Z_2)+(i_2+i_3)Z_2-U_1=0$$ $$\frac{i_2^3}{P_2}+i_2(Z_1+Z_2)+(i_1+i_3)Z_2-U_2=0$$ $$\frac{i_3^3}{P_3}+i_3(Z_1+Z_2)+(i_1+i_2)Z_2-U_3=0$$ ...
0
votes
1answer
68 views

number of solutions in homogeneous system

What is the maximum possible number of solutions of homogeneous system $N \times N$ ($N$ variables, $N$ equations) of degree $2$, where in each equation we have linear terms in $x_i$ and quadratic ...
3
votes
2answers
430 views

The system of equations $x^2 + y^2 - x - 2y = 0$ and $x + 2y = c$

I have $(1.) \quad x^2 + y^2 - x - 2y = 0 \\ (2.) \quad x + 2y = c$ Solving for $y$ in $(2.)$ gives $y = (c - x) / 2$ Is there a way to simplify equation $(1.)$? Because at the end I arrive at ...
3
votes
2answers
305 views

Solving 3 simultaneous cubic equations

I have three equations of the form: $$i_1^3L_1+i_1K+V_1+(i_2+i_3+C)Z_n=0$$ $$i_2^3L_2+i_2K+V_2+(i_1+i_3+C)Z_n=0$$ $$i_3^3L_3+i_3K+V_3+(i_1+i_2+C)Z_n=0$$ where $L_1,L_2,L_3,K,V_1,V_2,V_3,C$ and $Z_n$ ...
1
vote
3answers
113 views

Simultaneous solution(s) to $a^2+4b^2+4ab=0$ and $a^2+4b^2+32+16a-8b=0$?

Could you tell me just how should I solve this system: $$ a^2+4b^2+4ab=0\\ a^2+4b^2+32+16a-8b=0 $$ I can't remember the proceeding and it's driving me crazy. Thanks a lot