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
18 views

tridiagonal matrix with a corner entry from upper diagonal

I am trying a construct a matlab code such that it will solve an almost tridiagonal matrix. The input I want to put in is the main diagonal (a), the upper diagonal (b) and the lower diagonal and the ...
0
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2answers
26 views

Is it possible to have a system of equations that all equal 0, and not have each unknown's value be 0?

I'm doing about a 2 hour long homework assignment where by hand I must construct a 10x10 matrix representing a system of equations. Based on the pattern I'm seeing, I can tell all of the equations ...
10
votes
1answer
295 views

How prove this systems-equation has least two postive integers solution

Show that: for any $k\ge 100,(k\in N^{+})$, there exsit $p\in N^{+}$, such $$\begin{cases} a+b+c=k\\ abc=p\\ a>b>c \end{cases}$$ has at least two postive integers solution $(a,b,c)$ ...
1
vote
2answers
38 views

Solving simultaneous PDEs

Given the equations (1):$$\frac{\partial u}{\partial t}+g\frac{\partial \eta}{\partial x}=0$$ and (2):$$\frac{\partial\eta}{\partial t}+H\frac{\partial u}{\partial x}=0$$ can we combine the two ...
0
votes
1answer
23 views

Matrix with given row and column sums

Let $N$ and $K$ be two given integer numbers different from zero. Let $S_n$ with $n=1,...,N$ and $C_k$ with $k=1,...,K$ strictly positive integer numbers such that $$ ...
2
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4answers
51 views

Simultaneous Quadratic Equations: $x^2 + y ^ 2 - 2 x + 6y - 35 = 0$ and $2x + 3y = 5$

I've been given the task to simultaneously solve: $$x^2 + y ^ 2 - 2 x + 6y - 35 = 0$$ $$2x + 3y = 5$$ I've tried applying the substitution method by reordering the second equation to both $x$ and ...
1
vote
1answer
23 views

Solution of system of equations in prime fields

In 'Algebra', Artin writes that the system of equation: $$8x+3y = 3$$ $$2x+6y = -1$$ have no solutions in $\mathbb{F}_2$ and $\mathbb{F}_3$ as the determinant (of the coefficient matrix) evaluates ...
1
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1answer
38 views

System of ODEs obtained by using the method of characteristics for $u_x + 2u_t - 4u = e^{x+t}$

I have a question which requires me to use the method of characteristics in order to solve the PDE $u_x + 2u_t - 4u = e^{x+t}$. This results in the system of ODE's $\frac{dx}{dr} = 1 , \frac{dt}{dr} ...
0
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1answer
103 views

Solve the system $ x \lfloor y \rfloor = 7 $ and $ y \lfloor x \rfloor = 8 $.

Solve the following system for $ x,y \in \mathbb{R} $: \begin{align} x \lfloor y \rfloor & = 7, \\ y \lfloor x \rfloor & = 8. \end{align} It could be reducing to one variable, but it is ...
0
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2answers
36 views

Simple trigonometrical equations

I'm having difficulties in solving the simultaneous equations $$ \begin{cases} \sin(x+y)=\frac{1}{\sqrt{2}}\\ \cos(2x+y)=\frac12 \end{cases} $$ for $0^{\circ}\le x,y\le 90^{\circ}$. The answer is ...
1
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2answers
68 views

Analog clock with same hands - sometimes one can't tell time [duplicate]

There is an accurate analog clock, however both hands are the same size and shape. How many moments during a day a person can not conclude current time from the position of the hands? This is from a ...
0
votes
1answer
15 views

Differential system, a matrix with eigenvalue

Let's say that we have $n$ differential equations written in the form: $x'(t) = Ax(t) + v \exp(\lambda t)$, where $v$ is the eigenvector of $A$ such that $A v = \lambda v$ and $A$ is a $n \times n$ ...
0
votes
2answers
32 views

Can the following system be solved symbolically/analytically?

I have the following system of equations with variables $a,m$, and I'm wondering—can this system be solved symbolically/analytically? \begin{align} m &= 100 + \frac{ \left( 200 ...
2
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0answers
50 views

How to find whole number answers in systems of square root equations

Given the following 4 equations, can you find 4 whole number answers using whole number variable inputs? $x,y,z$ where $x>y>z$ $Eq 1 = (x^2-2xy+y^2-2xz+z^2)^{\frac{1}{2}} $ $Eq 2 = ...
0
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0answers
20 views

How do I solve massive system of equations (with lots of variables) quickly?

Just wondering how to solve system of equations involving 3+ unknowns quickly. In my math class, we're given questions like these which involve solving huge system of equations on a time limit, ...
0
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2answers
38 views

Find $x(t)$ and $y(t)$ which satisfy the following differential equations

Find $x(t)$ and $y(t)$ which satisfy $3\dot x + \dot y +5x-y=2e^{-t}+4e^{-3t}$, $\dot x + 4\dot y -2x+7y=-3e^{-t}+5e^{-3t}$, subject to $x=y=0$ at $t=0$. This is how I tried it: If we multiply ...
0
votes
2answers
142 views

$10$ Equations in $10$ variables

$x + y + z + u + v = 2$ $xp + yq + zr + us + vt = 3$ $xp^2 + yq^2 + zr^2 + us^2 + vt^2 = 16$ Similarly, $xp^3 + ... + vt^3 = 31$ Power $4,$ that is $xp^4 +... + vt^4 = 103$ Power $5 = 235$ Power ...
2
votes
3answers
59 views

Solve these equations simultaneously (trig)

Solve for $ x,y: $ \begin{equation}\cos x -\cos(x+y) = 0 \end{equation} \begin{equation}\cos y -\cos(x+y) = 0 \end{equation} The answers are $(0, 0), (\frac{2\pi}{3}, \frac{2\pi}{3})$. I get ...
0
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0answers
30 views

Solution of a general linear system of equations: 4-term n-equations

I have the following system of equations.... $$y_1 = c_{11} \cdot x_{11} + c_{12} \cdot x_{12} + c_{13} \cdot x_{13} + c_{14} \cdot x_{14}$$ $$y_2 = c_{21} \cdot x_{21} + c_{22} \cdot x_{22} + ...
0
votes
0answers
10 views

Can this equation have an explicit solution?

Given $n > 0$, $0 \leq i \leq n$ is an integer, $D = diag(d_1, \dots, d_n)$ is positive definite, $e_i$ is the $i$th column of a $n \times n$ identity matrix, $u \in R^n$ such that $B = D + u * ...
1
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0answers
35 views

Rank of a matrix with parameters

I have the following matrix: $$\begin{pmatrix} b+3 & a & 4 & -2b-1\\ b & -3 & 5 & -6\\ -1 & 1 & 2a+1 & 1-a \end{pmatrix}$$ How can I determine the rank for ...
0
votes
1answer
29 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
27 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
571 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
20 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 & & & & & & ...
0
votes
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
52 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
26 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
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0answers
35 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
votes
0answers
25 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
25 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
43 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?
0
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0answers
17 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
votes
0answers
20 views

Help Solving Trilateration Location Determination Example

I was reading about Trilateration on page 238 of this link: Trilateration Paper I pulled my equations from this paper. I made up some values for centers of 3 circles and an imaginary 'receiver' ...
0
votes
0answers
26 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
19 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
50 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
vote
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
86 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
27 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
0answers
34 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
54 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
29 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
21 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 ...