This tag indicates that several equations (of some type) must all hold. Do not use alone! Use in conjunction with (linear-algebra), (polynomials), (pde), (differential-equations), (inequalities) or another tag that describes the nature of the equations being considered.

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2
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1answer
15 views

Properties of the solution of a linear system with random equations

$x_i$ is drawn from $\mathrm{unif}(a,b)$, $y_i$ is drawn from $\mathrm{unif}(c,d)$. $x_i$ are independent from each other. $y_i$ are independent from each other. $x_i$ are independent of $y_i$. $i$ ...
1
vote
2answers
57 views

Basis for intersecting subspaces - is there a trick here?

I'm doing this problem, which gives me these subspaces of $\mathbb{R}^4$ $$U=\text{span}\left\{\;\begin{pmatrix} 3\\ 2\\4 \\ -1\end{pmatrix},\;\begin{pmatrix} 1\\ 2\\1 \\ ...
0
votes
2answers
44 views

Solution to system of non linear equations

what is the best way to solve this system of equations: $$ax^2 +by^2-2y=0$$ $$axy+byz-z=0$$ $$ay^2+bz^2-c=0$$ Solve for x,y,z where a,b,c are constants.
1
vote
0answers
61 views

System of Equations which can be solved by inequalities: $(x^3+y^3)(y^3+z^3)(z^3+x^3)=8$, $\frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}=\frac32$.

S367. Solve in positive real numbers the system of equations: \begin{gather*} (x^3+y^3)(y^3+z^3)(z^3+x^3)=8,\\ \frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}=\frac32. \end{gather*} Proposed by ...
0
votes
2answers
50 views

Determine $x$ if $x = 4 \mod 17$ and $x = 3 \mod 11$. [on hold]

Given $x =4\mod 17$ and $x = 3\mod 11$, determine $x$. I know that $\gcd(17,11)= 1$. I was hoping to use this to determine $x$.
0
votes
1answer
24 views

If $A$ is a singular square matrix, then $Ax = b \neq \vec{0}$ has $0$ or many solutions

I was reading this pdf: https://www.math.ohiou.edu/courses/math3600/lecture10.pdf and it tells you that if $A$ is a singular square matrix, then $Ax = b \neq \vec{0}$ has $0$ or many ...
1
vote
1answer
30 views

How can I write a set of equations in summation form?

I have a system of equations as follows: \begin{align} & A_1^{11} + A_1^{12} + A_1^{13} + \cdots + A_1^{1n}=X \\[8pt] & A_1^{21} + A_1^{22} + A_1^{23} + \cdots+ A_1^{2n}=X \\[8pt] & ...
0
votes
0answers
15 views

Counting the number of roots of multivariate polynomials?

The equation of a circle is well known $$(x-x_0)^2+(y-y_0)^2 - r^2 = 0$$ It has a solution all along the circle with midpoint $(x,y) = (x_0,y_0)$. We also know that $ab = 0$ whenever any of $a$ and/or ...
0
votes
1answer
27 views

solving a pair of simultaneous equations

I have a rather messy pair of simultaneous equations, which I need to solve for x: ...
1
vote
2answers
37 views

So many logs with different bases

$ \large { 6 }^{ \log _{ 5 }{ x } }\log _{ 3 }( { x }^{ 5 } ) -{ 5 }^{ \log _{ 6 }{ 6x } }\log _{ 3 }{ \frac { x }{ 3 } } ={ 6 }^{ \log _{ 5 }{ 5x } }-{ 5 }^{ \log _{ 6 }{ x } }$ The sum of ...
1
vote
2answers
41 views

Number of integral solutions for an equation

How do we approach this kind of problem of finding number of positive integral solutions to $$\frac{1}{x}+\frac{1}{y} = \frac{1}{n!}$$ Here $n$ is given.
0
votes
1answer
6 views

How is Dulac's Multiplier selected?

I'm aware that Dulac's (Negative) Criterion states that a system of differential equations of the form $x' = f(x,y), \; y' = g(x,y)$ has no periodic orbits in the plane if we can find some function ...
0
votes
0answers
20 views

Finding the General Solution for a System of Differential Equations with Complex Eigenvalues

I think I might just be having trouble with formatting my answer, because I'm fairly sure my work is right up until this point. The question asks to find the general solution to $$X'= ...
0
votes
0answers
15 views

Euler explicit and semi-implicit

I am given a simple dynamic system with an initial condition: $a(t) = 0.9 - 0.1v(t)$ $v(0) = x(0) = 0$ I want to calculate $x(1)$ with a time step of $\Delta t = 1$ using Euler explicit and semi ...
0
votes
0answers
18 views

Nonlinear equations algorithm - Newton method

Some time ago I posted a question regarding the simple case of finding the intersection point when I have only two functions, and with your help I found an answer. It was this case: $f(x) = a + ...
2
votes
1answer
73 views

A seemingly-trivial divisibility conjecture

While working on another problem, I stumbled on the following divisibility claim. Conjecture: No integers $a,b,c,d$ satisfy all of the following conditions: $a^2+b^2-c^2-d^2 = 2(ad-bc)-1$; ...
1
vote
2answers
27 views

Solving system of non-linear equations.

So I'm trying to find the stationary points for $$f(x,y,z) = 4x^2 + y^2 +2z^2 -8xyz$$ Setting the partial derivatives to zero leads to: $$x-yz=0 \\ y-4xz=0\\z-2xy=0$$ Substiting $z=2xy$ into the ...
0
votes
3answers
40 views

Real problems solved with systems

Can anybody tell me where can I find some REAL problems (i.e. form real life) that can be solved using a 3x3 system of linear equations? Or, can anybody give me an example? A solution could be a ...
1
vote
1answer
25 views

Nonlinear equations systems

Can anybody help me to find a system with 3 equations and 3 unknowns and a bounded domain D = [a,b]x[c,d]x[e,f] such that the system has an unique solution in D? Also, i need nice equations, because ...
-3
votes
0answers
35 views

Advance Algebra [closed]

I have an interesting problem if anyone can solve it: Harry rides by train to go home and arrives at the railway station every day at 18h00. His wife travels by car to pick him up at the station and ...
3
votes
2answers
50 views

The condition about some positive real numbers can be written as the sum the nearby two

Given $n$ positive real numbers $x_1,...,x_n$. What is the condition that they can be written as $$x_1=y_1+y_2$$ $$x_2=y_2+y_3$$ $$\ldots$$ $$x_n=y_n+y_1$$ where $y_1,\ldots,y_n$ are also some ...
0
votes
1answer
68 views

System of equation $x+y+z=2007; xyz=14000$

I have to solve the system of equations $$\begin{cases} x+y+z=2008,\\ xyz=14000, \end{cases}$$ where $x,y,z$ are positive integers such that $1\le x \le y \le z \le 2000.$ My work so far: ...
0
votes
1answer
24 views

Consistency of system of linear equations

Find when the equations $$\begin{cases}x + y - 2z = 0\\ax + by + cz = 0\\bx + cy + az = d\end{cases}$$ are consistent and solve them completely when they are consistent. I have tried the ...
0
votes
0answers
43 views

Solution of Nonlinear System of Equations

I wand to find the solutions $p_H, p_L$ implied by the following two equations: (I) $$\frac{(1-\lambda)(p_L-c_L)}{p_H \frac{q_L}{q_H}-c_L} = \frac{\lambda(p_L-c_L)}{p_H-(q_H-q_L)-c_L} - \lambda$$ ...
0
votes
0answers
24 views

How to Solve this Nonlinear System of Equations?

There are 24 variables and 24 equations in the system: $ i=0,1,2,3\\ \textrm{variables}: s_i,\ t_i,\ a_m \; (m=0,...,15)\\ \textrm{constants}: b_{ni} \; (n=0,...,5)\\$ $$\begin{array}{rcl} a_0\cdot ...
2
votes
6answers
124 views

Solve the system of equations: $a+b+c=2$, $a^2+b^2+c^2=6$, $a^3+b^3+c^3=8$ [closed]

If we have \begin{cases} a+b+c=2 \\ a^2+b^2+c^2=6 \\ a^3+b^3+c^3=8\end{cases} then what is the value of $a,b,c$?
1
vote
2answers
69 views

The sum of two numbers is 5/9…

The sum of two numbers is $\frac{5}{9}$. The quotient of the two numbers is $1$. What is the product of $40$% of each number? The answer I got was $\frac{1}{81}$. I don't understand this - would ...
0
votes
0answers
29 views

What are the eigenvalues of the following Hermitian matrix?

Let $\mathtt{i}=\sqrt{-1}$ and $$p=1+\mathtt{i}=\bar{q},\ \ q=1-\mathtt{i}=\bar{p}.$$ Let $A$ be an $n\times n$ matrix such that $$A=\begin{bmatrix} 0 & p & p & \cdots & p & ...
0
votes
1answer
19 views

how do you solve the simultaneous equations 2x + y = 9 and x - 2y = -8 using the matrix method?

i first convert the bases into a 2x2 matrix and then i multiplied the inverse matrix by the 2x1 e/f matrix of 9 (on top) and -8 (on the bottom) which gave me x = 5.4 and y = 5 however the answer ...
0
votes
1answer
24 views

Find the length of the longest diagonal of the bo

The total length of all $12$ sides of a rectangular box is $60$. The surface area of the box is given to be $56$. Find $(i)$ the length of the longest diagonal of the box $(ii)$ the volume of the box ...
3
votes
1answer
47 views

Symmetric system of equations problem

Solve the following simultaneous eqations on the set of real numbers: $$a^2+b^3=a+1$$ $$b^2+a^3=b+1$$ I have found two trivial solutions: $$a=b=1$$ $$a=b=-1$$ but I can't prove that there are no ...
0
votes
0answers
21 views

How to numerically minimize system of equations composed of data and smoothness terms, ensuring minimum solution norm

I need to find $g$ that minimizes: $$\sum_{v=0}^n (f+g_{v_{left}}-g_{v_{right}})^2 + \frac{1}{\lambda}\sum_{v=0}^m (g_{v_i}-g_{v_j})^2$$ where $f$ is constant and the sums are over pair of $v$ ...
1
vote
0answers
21 views

Reducing a system to first order

Convert the following to a first order system $$x''(t) = k_x(x(t) - y(t))^{-2}, \ \ y''(t) = k_y(x(t) - y(t))^{-2},$$ $$x'(0)=v_x, \ \ y'(0) = v_y, \ \ x(0) = x_0, \ \ y(0) = y_0.$$ I know how to ...
0
votes
0answers
8 views

Closed-form solution for system of equations for finding a critical point

I am trying to find a critical point of a function $\mathbb{R}^d \to \mathbb{R}$ by setting its gradient to zero. I would like to solve the follwoing system of equations. $$\frac{1}{1 - \sum_{j=1}^d ...
1
vote
1answer
35 views

Systems of equation

Find non-negative solutions of systems of equations: $$\begin{cases} x^2y^2+1=x^2+xy \\ y^2z^2+1=y^2+yz \\ z^2x^2+1=z^2+zx \end{cases} $$ My work so far: 1) $(1;1;1) - $ solution. 2) ...
1
vote
3answers
63 views

What does x equivalent to 2 mod 15 mean?

I came across the following question: Consider the following system of equivalences of integers. $$ x \equiv 2 \bmod{15} $$ $$ x \equiv 4 \bmod{21} $$ The number of solutions in $x$, where $1\le ...
1
vote
1answer
36 views

Solve $A^kx=b$ system using $LU$

I have the system $A^kx=b$ and the $LU$ factorization $A=LU$. How can I solve the system without actually calculating $A^k$?
1
vote
1answer
97 views

Solve this systems to condition $3x^3(x+1)^2=2y^2(z+3)^3$

if $x,y,z$be postive real numbers, solve systems of this following equation $$ 3x^3(x+1)^2=2y^2(z+3)^3\tag{1}$$ $$3y^3(y+2)^2=2z^2(x+1)^3\tag{2}$$ $$3z^3(z+3)^2=2x^2(y+2)^3\tag{3}$$ My approach is ...
4
votes
2answers
64 views

Why is a polynomial $f(x)$ sum of squares if $f(x)>0 $ for all real values of $x$?

If a polynomial $f(x)>0 $ for all real values of $x$, then $f(x)$ is sum of squares. Why is this true ? I understand that the roots of this $f(x)$ will be complex and hence will exist as ...
0
votes
0answers
13 views

Sum of triangular matrices system

I was wondering if there is a nice way to solve the following linear system of equaitons: $(A+B) x = b$, where $A$ is an upper-right triangular matrix (all elements higher than the main diagonal are ...
-1
votes
1answer
27 views

Rewriting system as a set of first order equations.

What I'm given: $$x'' = x' + y' + x + y$$ $$y'' = 2x' + 3y' + 3x + y$$ $$z=x'$$ $$w=y'$$ My solution: We know that $z'=x''$ and $w'=y''$. We can write: $$z'=z+w+x+y$$ $$w'=2z+3w+3x+y$$ I'm not ...
1
vote
0answers
36 views

Resolve integral equations

There is a way to solve this problem? Let be $[a,b]$ an interval where $a$ is finite but $b$ can also be infinity. Find a function or a distribution $h(u,s)$ for $u,s \in \mathbb{R}$ such that for ...
0
votes
0answers
9 views

When does a system of n symmetric polynomials in n variables have exactly one solution over C up to permutation?

I was slightly amused that if I never learned about polynomials and was asked if Vieta's system of equations has exactly one solution up to permutation, the solution would be to develop polynomials in ...
1
vote
1answer
23 views

Analytical solution of a partial system of differential equations

Consider the following system of PDEs: $$\left\{ \matrix{ {{\partial f} \over {\partial y}} + {{\partial g} \over {\partial z}} = - \left( {8x + 5z} \right) \hfill \cr {{\partial f} \over ...
0
votes
1answer
24 views

How do I solve this 3-D system of linear equations using Gaussian elimination?

I have the following system of equations: $x+2y+3z = -6$ $2x - 3y - 4z = 15$ $3x + 4y + 5z = -8$ I came up with this: $x + 2y + 3z = -6$ $-7y - 10z = 18$ $5x + y + z = 7$ Can you tell me the ...
0
votes
0answers
34 views

First-order system of linear differential equations [Revision]

$$\frac{dx}{dt}+y=0 \quad \text{and} \quad \frac{dy}{dt}-x+2y=\sinh{t}$$ (Oxford, 2011) First, we isolate $y$ from the first equation, $$y=-\frac{dx}{dt} \implies ...
2
votes
1answer
27 views

Solving $n$-binary-variable system of equations using only combinations of $n \over 2$ variables when $n \over 2$ is even

It seems that it's impossible to find the unique solution to an $n$-binary-variable system of XOR equations if you only use all $(n \text{ choose } {n \over 2})$ equations combining half the ...
0
votes
1answer
62 views

Solving system of eqations

Find All $(x,y,z) \in R$ such that : $\begin{align*} x^2+y^2+xy &= 37 \\ x^2+z^2+zx &= 28 \\ y^2+z^2+yz &= 19 \end{align*}$ My approach is as follows: I noticed that these expressions ...
-2
votes
1answer
71 views

Find the fixed points of the system, and sketch the trajectories of the system

I am given the following system: $$x' = [(x-1)^2 + y^2]y$$ $$y' = -[(x-1)^2 + y^2]x \tag{*}$$ where $x = x(t), y = y(t)$. I am supposed to Find the fixed points of the system, and ...
1
vote
0answers
23 views

System of Nonlinear Equations (sum of powers)

I want to show the only solution to the following system of equations is the trivial one ($x_{i} = 0$). I don't know if this is true, but I think it should be. Let $x_{i} \in \mathbb{C}$ for $1 \le i ...