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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1answer
34 views

System of linear equations where unknowns can only be +1 or -1

I have a system of linear equations, in which the unknowns can only take 2 integer values: +1 or -1. The linear system is $$ Ax = 0 $$ Matrix A is shown below with dimension (3 x 14): $$ ...
2
votes
0answers
29 views

How to solve the equation $Au+Bv=C$

How do I solve $Au+Bv=C$ Where $A$ and $B$ are constant known matrices that are nxn, $C$ is a constant known nx1 vector while $u$ and $v$ are unknown nx1 vectors with the condition given that $u_i = ...
3
votes
3answers
591 views

Question about a solution of a system of three non linear equations in three unknowns

Let $a$, $b$ and $c$ be positive real numbers such that $$ a + \frac{1}{b} = 3$$ $$b + \frac{1}{c} = 4$$ $$ c + \frac{1}{a} = \frac{9}{11} $$ then $$ a \times b \times c =?$$ I tried doing this ...
1
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0answers
24 views

How to diagonalise this pentadiagonal pseudo-Toeplitz matrix?

How can one diagonalise this N-by-N pentadiagonal matrix (where $r$ is some real constant)? $$ \tiny \begin{pmatrix} r^2 +r & -2r -1 & 1 & & & & & & ...
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0answers
35 views

Maximum of $x_1 - x_2 - x_3 + x_4 - 2x_5$ with some constraints

I have to find maximum of $x_1 - x_2 - x_3 + x_4 - 2x_5$ with constraints: $-x_1 +x_2 + x_3 = 2$ $x_1 + 2x_2 + x_4 = 10$ $x_1 - x_2 + x_5 = 4$ of course $x_i \ge 0$. From constrains I have: ...
1
vote
3answers
54 views

System of equations with radicals

Solve the system of equations (in $\mathbb R$): $$\begin{matrix} 2\sqrt[4]{\frac{x^4}{3}+4}=1+\sqrt{\frac{3}{2}y^2} \\ 2\sqrt[4]{\frac{y^4}{3}+4} = 1+\sqrt{\frac{3}{2}x^2} \end{matrix}.$$ This ...
0
votes
1answer
27 views

How can I solve this system of linear different equations?

Here's the system $$\frac d{dt} \begin{bmatrix} x \\ y \\ z \\ p_1 \\ p_2 \\ p_3\end{bmatrix} = \begin{bmatrix} 0 & A \\ B & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ p_1 \\ p_2 \\ ...
0
votes
0answers
44 views

Classification of critical points for plane autonomous system

Okay so I've changed the 2nd order nonlinear ODE $$ x'' = a(x')^2 - ax' -ax $$ where a is a real constant, into $$ x' = y $$ $$ y' = ay^2 -ay - ax $$ I'm asked to verify the critical point (0,0). ...
0
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0answers
26 views

Solution to a ODE system using a power series

I'm certain the pattern the system creates is $$ A^kX(0) = \begin{pmatrix}2^k\\1\\2^k\end{pmatrix}\hspace{3pc} $$ Where A is a matrix created by the system and X(0) is a solution vector at t=0 Im ...
0
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0answers
27 views

How many solutions does this boolean equation system has?

How many solutions does this boolean equation system has? $$\left\{ ...
0
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0answers
28 views

A few questions about eignenvectors and the associated root vectors.

Let A be the matrix formed from the original system of equations and t is a repeated eigenvalue. I've noticed when solving problems containing eigenvectors of multiplicty >1 that when the ...
1
vote
1answer
32 views

System of linear equations with four unkowns

I have no idea how to solve this system of equation : $$\begin{align}u+v+w&=7 \\v+w+x&=-8 \\w+x+u&=5 \\x+u+v&=-10\end{align}$$ I usually use the addition/substraction method, but ...
0
votes
2answers
19 views

A simultaneous equation question

$38$ bottles of soda was consumed by $18$ women. Some took $2$ and others took $3$ . (A) How many women took $2$ sodas? (B) How many women took $3$ sodas? I thought I might use simultaneous equations ...
1
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2answers
46 views

Question on power, If 2x^2x^2x^2x… =4 Solve for x

I've seen this random example, in which can anyone give me clue how to solve for $ x $ here?
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0answers
19 views

Stuck on polynomial equation in optimization problem

I've been trying to solve an optimization problem, but I am completely stock on one step. I had the following Langrangian: $$\nabla\mathcal{L}(x,\lambda)= e\frac{\sum_{t\in I}e^t \Delta P(t)( x^t ...
0
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0answers
27 views

Identifying a sequence of numbers from an optimization problem in $L^1$

Question Does there exist general closed form solutions (or some sort of recurrence relation) to the system of equations: $$\begin{align} x_0 &= -1\\ x_{k+1} &= 1\\ \sum_{j = 0}^k (-1)^j ...
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0answers
21 views

Constrained System of Equations

$Ax=b$ is a linear system of equations with dimension of $n$ in which $A$ is real, symmetric, and positive definite (RSPD). The matrix $A$ can be also written as \begin{equation} A_{n \times n} = G_{n ...
0
votes
1answer
19 views

Unique solution of nolinear equation set

$$\left\{ \begin{aligned} f_1(x_1,x_2...x_n)=0 \\ f_2(x_1,x_2...x_n)=0 \\ \vdots \\ f_n(x_1,x_2...x_n)=0 \end{aligned} \right. $$ $f_i\in C^\infty(R^n)$,what is the condition that make the equation ...
0
votes
1answer
31 views

Interpreting matrices as linear equations.

$$ \left[ \begin{array}{cc|c} 3&4&5\\ 4&5&4 \end{array} \right] $$ Could I say that this augmented matrix forms two different planes ($3x_1 + 4x_2 = 5$ and $4x_1 ...
1
vote
2answers
52 views

How do I Solve this Seemingly Simple Set of Four Equations with Four Unknowns?

I have what looks like a set of simple simultaneous equations: 4 equations with 4 unknowns. The numbers are really simple, and in fact I already know the answer, but I cannot figure out how to work ...
1
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0answers
62 views

System of (non linear) equations

Let $n \geq 2$. Could it be proved that the following system, with $z_k\in \mathbb C$, $ \begin{cases} z_1^n + z_{n}z_1^{n-1} + z_{n-1}z_1^{n-2} + \cdots + z_2z_1+z_1 & = 0 \\ z_2^n + ...
4
votes
2answers
94 views

Any methods of solving this system of ODE's?

I try to solve this system of ODE's: $$ \frac{dQ_1 (t)}{dt} = - a \sin (\omega t) Q_2(t) + b \cos(\omega t) Q_3(t) $$ $$ \frac{dQ_2 (t)}{dt} = - a \sin (\omega t) Q_1 (t) - c Q_3(t) $$ $$ ...
0
votes
0answers
31 views

closed form or approximate solution for a system of equation : $m(t)=v\sin(\arctan(at+b))+v\sin(\arctan(ct+d))$

Can one solve for $(v,a,b,c,d)$ the following equation ? $t$ takes discrete values and $m(t)$ is known for as many $t$ needed. However please assume that special values of $t$ may not be available ...
3
votes
2answers
58 views

Shamir's secret sharing interpolation problem

I try to understand this protocol - Shamir's secret sharing - threshold scheme. I got my data and I made interpolation basing on examples published on Wikipedia. You can see them below (sorry, I am ...
1
vote
2answers
68 views

Write down a homogeneous linear system of three distinct equations in three variables that has the non-trivial solution $(x, y,z) = (1,2, 4)$

Write down a homogeneous linear system of three distinct equations in three variables that has the non-trivial solution $(x, y, z) = (1,2, 4)$. I am confused on how to approach this problem
3
votes
1answer
32 views

Sums of Pairs of Integers

Suppose that $a,b,c,d,e$ are integers with the constraint that $a\leq b \leq c \leq d \leq e$. Also, suppose that the sums of the $5 \choose 2$ = $10$ pairs (i.e. $a+b$, $a+c$, $a+d$, $b+c$, $b+d$, ...
0
votes
0answers
8 views

Solutions of $\sum_{n=1}^N a_n n\sin{(n x+\theta_n)}=\sum_{n=1}^N a_n n^2\cos{(n x+\theta_n)}=0$

Is there a solution for the equation $\sum_{n=1}^N a_n n\sin{(n x+\theta_n)}=\sum_{n=1}^N a_n n^2\cos{(n x+\theta_n)}=0$ in terms of the variable $x$, for some choice of coefficients $a_n$ and ...
1
vote
1answer
24 views

Introducing noise and time lag between two coupled Rössler systems

I have two Rössler systems mutually coupled by the second component. I want to introduce some small noise and a slight time lag of the coupling between the systems. I'm not sure 1. what the best ...
0
votes
4answers
59 views

If $ax + by = a(b-1) + b(-1)$, then does $x = b-1$ and $y = -1$

In this case, $x$ and $y$ are variables and $a$ and $b$ are arbitrary constants. It seems like just looking at the equation that this would be true, but is there a case when it does not work? If I try ...
1
vote
1answer
40 views

Is there a general coordinate transformation perserving the components of an Euclidean metric?

In the Euclidean space (or Lorentz spacetime, if you are interested in relativity), there is one orthonormal coordinate system $\{x^\mu\}$ such that the distance squared is given by ...
1
vote
2answers
40 views

How to prove that equation over probabilities has unique solultion or find counterexample?

Given equations: $$ \prod_{i=1}^n p_i = \prod_{i=1}^n (1-p_i)= \frac{1}{2^n} $$ where $p_i\in (0,1), i=\overline{1,n}$. Is it true that this system has unique solution $p_1=p_2=\ldots=p_n=\frac12$ ...
3
votes
1answer
81 views

Proving the equations $x_1+\dots+x_n=0$, …, $x_1^n+\dots+x_n^n=0$ have a unique solution

Let equations of the form $\left\{\begin{matrix} x_{1}+x_{2}+...+x_{n}=0\\ x^{2}_{1}+x^{2}_{2}+...+x^{2}_{n}=0\\ .........\\ x^{n}_{1}+x^{n}_{2}+...+x^{n}_{n}=0 \end{matrix}\right.$. Proof: ...
0
votes
1answer
34 views

Help in understanding step function calculation

Dear community I would appreciate if you can help me understand these equations. I mean how did he jump from line 1 to line 2? How do u[n] get cancel? Then in the last line where did the "8" come ...
-1
votes
1answer
24 views

Question regarding arbitrary parameters

Solve the following system of linear equations: x + y + z = 4 x + y + z = 4 2x + 2y + 2z = 8 I'd like some help understanding how to go about solving this. I ...
0
votes
1answer
22 views

Consequence of linear combination in matrix .

If a column of a matrix is linear combination of another column, what are the consequences ? Several terminology coming into my mind to relate with this such as Rank of the matrix ; Determinant ...
2
votes
1answer
48 views

Two variable equation

I'm stuck with the following example (42.). Some help is much appreciated. Thank you.
1
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0answers
45 views

Can trigonometric equations be graphed?

I was solving various trigonometric equations. I was confused that how are they solved easily by using methods that are useful to solve algebraic equations. Do the trigonometric functions in ...
0
votes
3answers
40 views

Solve the Equation.

$$ \begin{bmatrix} 1 & 1 \\ 1 & 1 \\ \end{bmatrix} \begin{Bmatrix} v_1 \\ v_2 \\ \end{Bmatrix}= \begin{Bmatrix} 0 \\ 0 ...
0
votes
1answer
62 views

System of equations and the Brouwer's Fixed-Point Theorem.

Let's consider the following system of equations: \begin{eqnarray}{ e^{xyz} = \frac{x}{\sqrt{e^{2xyz}+1}}\\ \cos(x+y+z) = \frac{y}{\sqrt{e^{2xyz}+1}}\\ \sin(x+y+z) = \frac{z}{\sqrt{e^{2xyz}+1}} ...
0
votes
0answers
20 views

Limit of solution equal to solution at limit

I have a system of equations (not DE, think algebraic equations) that depends on a parameter. I am trying to learn under what conditions on the system is a limit (as the parameter converges to some ...
2
votes
1answer
56 views

Finding solutions to a system of linear equations

I was working on this problem, to which I have the answer, but it is just that, an answer with no explanation, and I am stuck on how the answer was arrived at, for a few parts of this question. In ...
0
votes
1answer
20 views

Apply gauss method to a linear system and them use results in another system

I have an exercise for my last assignment of linear algebra, which is the following: I tried to row reduce to echelon form the matrix created by the first linear system of equations and I obtain ...
0
votes
2answers
18 views

Don't understand adding a system of compound inequalities

I'm reading a proof of the Division Theorem and one line that comes up is Since 0 ≤ r1 < b and 0 ≤ r2 < b , we have −b < r1 − r2 < b. I do not ...
2
votes
3answers
189 views

Find $p$ for which all solutions of system/equation are real

There is system of $5$ equations $$ a+b+c+d+e = p; \\ a^2+b^2+c^2+d^2+e^2 = p; \\ a^3+b^3+c^3+d^3+e^3 = p; \\ a^4+b^4+c^4+d^4+e^4 = p; \\ a^5+b^5+c^5+d^5+e^5 = p, \\ \tag{1} $$ where $p\in\mathbb{R}$. ...
0
votes
1answer
31 views

simultaneous equations-exponential and linear

I am trying to find a general formula for x and y given that $y=mx+c$ and $y=Ae^{kx}$, with m, c, A and k as constants (and e is Euler's number). essencially, find the point(s) where an exponential ...
0
votes
1answer
26 views

System of equations with 3 variables - Does order matter?

I'm studying Algebra and I'm now at topic 'System of equations with 3 variables'. I'm having a hard time with the following example: $$ \begin{cases} 2x + 2y + 3z = 10\\ 3x + y-z = 0\\ x + ...
1
vote
2answers
303 views

Matrix Derivations-Research

First off, I know that this problem is extremely long and can be intimidating. I have put a lot of work and thought into it. In reference to what loup blanc has responded with, I am not dismissing ...
2
votes
0answers
142 views

Nonnegative solution of a linear system

Given three collections of parameters $\epsilon_1 > ... > \epsilon_N$, $(a_1,...,a_{N-1})$ and $(b_1,...,b_N)$ that satisfy the following conditions $\forall i, a_i \geq 0, ...
0
votes
1answer
62 views

How to solve over-determined linear system of equations?

at the moment I am working on an application where I have to solve some systems of linear equations during the whole algorithm. Because the programming-language I have to use is something related to ...
0
votes
2answers
116 views

System of $24$ variables

Assume that $a_1, a_2,\ldots, a_{24}$ satisfy $$a_1+a_2+\ldots+a_{24}=26$$$$a_1^2+a_2^2+\ldots+a_{24}^2=26$$$$\vdots$$$$a_1^{24}+a_2^{24}+\ldots+a_{24}^{24}=26$$ Find $a_1a_2⋯a_{24}$. How do I solve ...