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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2answers
36 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
115 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 ...
1
vote
3answers
54 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
25 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
9 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
28 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
22 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 = ...
-1
votes
0answers
16 views

Number of escalator steps we can see [on hold]

A man walks up an escalator that moves up and counts 50 steps. The next day he walks up the same escalator and counts 75 steps. If the second speed (in steps per time unit) is three times the first ...
3
votes
3answers
543 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
vote
0answers
18 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
32 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
38 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
24 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
21 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
24 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
27 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 ...
0
votes
1answer
25 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 ...
1
vote
1answer
51 views

How I can solve this exercice? [closed]

We want to divide \$34800 over a group of people with equality. If there are 5 absent persons, The amount of each person of this group will increase to \$1160. How many of those persons are ...
0
votes
2answers
10 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 though I ...
1
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2answers
42 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
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
26 views

Perfect equilibrium - consumer, producer surplus

Inverse function of market demand for certain good is equal to $P=100-0.25Q$, inverse supply function is $P=20+0.55Q$. Calculate equilibrium price and quantity. Furthermore calculate consumer and ...
0
votes
0answers
25 views

Perfect competition - question about profits [closed]

Market is supplied by: 50 competitive companies all of them have relatively low costs given by an equation $C_l(q)=350+2q+q^2$ and by n companies of higher costs given by an equation ...
0
votes
0answers
7 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 ...
0
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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
30 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
49 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
votes
1answer
43 views

How to solve an equation with 2 variables? [closed]

For example, the equation $\sqrt{2(x-y)^2+6x-2y+4}=\sqrt{y} + \sqrt{x+1}$ has the root of $y=x+1$, or the equation $(4x^2+1)x+(y-3)\sqrt{5-2y}=0$ has the root of $2x=\sqrt{5-2y}$. What do these ...
1
vote
0answers
52 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
82 views

Any methods of solving this system of ODE's?

I try to solve this system of ODE's: $$ \frac{dQ_1 (t)}{dt} = - a \sin (\omega t) Q_2(t) + b \cos(\omega t) Q_3(t) $$ $$ \frac{dQ_2 (t)}{dt} = - a \sin (\omega t) Q_1 (t) - c Q_3(t) $$ $$ ...
1
vote
0answers
22 views

Can someone explain what independent linear equations are?

Can someone explain what independent linear equations are? Specifically whether the following equations are independent, or even linear equations? $$\frac Y{X-1}=\frac XY$$ $$Y=\left(\frac ...
0
votes
0answers
21 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
28 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
49 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
2
votes
1answer
21 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
6 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
17 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
54 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
34 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
38 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
78 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
32 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
21 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 ...
0
votes
2answers
24 views

Find a line parallel to a known line that intersects a known circle at one point.

There is a circle with an equation $x^2+y^2=16$ and a line with equation $y=x+1 $. The question is to find an equation of line placed parallel to this line and touching the circle at only one point. ...
2
votes
1answer
43 views

Two variable equation

I'm stuck with the following example (42.). Some help is much appreciated. Thank you.