Questions tagged [systems-of-equations]
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.
7,956
questions
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26
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How do we solve this system of two trigonometric equations?
Given a system of 2 equations:
$\arccos(f_1(x))=\arccos(g_1(y))$
$\sin(f_2(x))=\sin(f_3(x))$
where:
Q1: $f_1(x), f_2(x), f_3(x),g_1(y)$ are linear equations over $x$ and $y$ respectively, with ...
- 41
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votes
0
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10
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Finding an $\mathbb{F}_q$-point on one specific intersection of quadrics
Let $\mathbb{F}_q$ be a finite field of large characteristic and $a_1, a_2, \cdots, a_n \in \mathbb{F}_q$ be some pairwise different elements. I assume that $\sqrt{-1} \in \mathbb{F}_q$. Consider the ...
- 243
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12
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Could distortion parameters of an image be retrieved by measuring lines only?
i have the following problem: i am given an image of an ordinary camera which is distorted meaning straight lines are not mapped into straight lines, but rather in "curvy" lines. My goal is ...
0
votes
0
answers
25
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what is the full formulation of this expression?
I have this expression that i am confused about:
$ \sum_{\forall i \in I(a)} D_{j,i}.x_i \geq 1 $ , $ \forall C_j \in J(a) $
Does it equal to the following one:
$ \sum_{i \in I(a)} D_{1,i}.x_i \geq ...
- 25
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votes
0
answers
20
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Fitting a sine function
Consider the following system over a periodic array, where $1\leq i,j\leq n$,
$$
T_j= \sum_{k=0}^n \frac{e^{-\sum_{|i|\leq k}(k-|i|)f_{j+i}/v}-e^{-\sum_{|i|\leq k}(k+1-|i|)f_{j+i}/v}}{\sum_{|i|\leq k}...
- 3,027
1
vote
1
answer
38
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Constructing a parallelepiped with all faces having equal diagonals, what goes wrong?
I'm trying to construct a parallelepiped where each face has diagonal lengths $d_1$ and $d_2$. I start by constructing a parallelogram in the $x,y$ -plane that has the correct diagonals. I put that it ...
- 7,686
1
vote
1
answer
75
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Help with Linear Algebra Questions
I'm currently studying linear algebra and I'm struggling with a few questions. I would appreciate any help or guidance you can provide.
Question 1:
I need to show that the line
\begin{align}
x-2=\frac{...
- 117
3
votes
2
answers
94
views
Solve the system : $ x^2 + x - 1 = y $, $ y^2 + y - 1 = z $, $ z^2 + z - 1 = x $ .
Solve the system of equations in x,y,z : $ x^2 + x - 1 = y $ , $ y^2 + y - 1 = z $, $ z^2 + z - 1 = x $ .
I'd tried analysing various manipulations but can't figure out .
What I've tried:
$$ x^2 + x - ...
- 67
0
votes
1
answer
55
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Optimal portfolio weight derivation.
I cannot see how to derive the following optimal portfolio weight $w^*$:
$$
w^*
=
\frac
{(b - \frac{1}{2}) \rho \sigma_1\sigma_2 - b\sigma_2^2 +\sigma_1\sigma_2\sqrt{(b - \frac{1}{2})^2\rho^2 + b(1-b)}...
- 101
1
vote
1
answer
69
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When does a system of linear equations have no solution?
I've just finished the first lecture of MIT 18.06 Linear Algebra with Gilbert Strang. The professor briefly discusses how one can find out whether $A x = b$ as a solution.
I don't understand this ...
- 23
2
votes
2
answers
134
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Prove: if $\bf{AB^T}$ is skew-symmetric and $\bf A$ full-rank, then $\bf{AX}=\bf B$ has unique solution $\bf X$
I've run into this statement while trying to prove that the energy of a rotating body in $N$ dimensions is conserved, it's the last puzzle piece I'm missing.
Let $\bf A$ and $\bf B$ be two $M \times N$...
- 215
1
vote
1
answer
18
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three-compartment model, analytic solution
I'm trying to solve the following problem. Given the following system of differential equations
$\frac{dN_1}{dt}=-\lambda_1 \cdot N_1$
$\frac{dN_2}{dt}=\lambda_1 \cdot N_1 -\lambda_2 \cdot N_2$
$\frac{...
- 45
2
votes
1
answer
79
views
Solve $\sin a =\sin x+\sin y, \cos a =\cos x+\cos y$ for $x$ and $y$ in terms of $a$
I saw this on quora.
Solve
$\begin{array}\\
\sin a
&=\sin x+\sin y\\
\cos a
&=\cos x+\cos y\\
\end{array}
$
for $x$ and $y$
in terms of
$a$.
Here is my solution.
My question:
is there a better ...
- 104k
0
votes
2
answers
46
views
Is there any tool or technique that allows solving a system of 8 quadratic equations?
I'm looking for a tool, or at least technique, that allows me to solve the following system of equations:
...
- 1,157
1
vote
1
answer
64
views
How to solve this system of multivariate polynomial equations for $0<x_7<x_6<x_8 \le 1$? Groebner basis maybe?
I am reformulating my question according to the guidelines I was given.
I have the following problem: I cannot find a way to solve the system of equations further down. This is the calculations from ...
- 11
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0
answers
21
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how to multiply two discrete signal function?
enter image description here
enter image description here
enter image description here
enter image description here
Is it not just termwise multiplication ?
I do not know how I get them.
Any answer ?
- 1
1
vote
2
answers
101
views
How to find the closed solution for \begin{cases} a^2+bc=-d\\ d^2+bc =-a\\ b(a+d)=b\\ c(a+d)=c\\ \end{cases}?
Given the following equations.
\begin{cases}
a^2+bc=-d\\
d^2+bc =-a\\
b(a+d)=b\\
c(a+d)=c\\
\end{cases}
My attempt:
Eliminating $bc$ in the first two equation, I have
\begin{align}
a^2-d^2 &= a -d\...
0
votes
2
answers
74
views
How to find roots of a system of multivariate polynomials? [closed]
I am trying to find the roots of a system of 3 multivariate polynomials with 3 variables. The polynomials are really 'ugly'. So far I have tried to find a Groebner Basis in Maple and got a Groebner ...
- 11
2
votes
1
answer
58
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Does this system of equations have a solution? [closed]
Over $\mathbb{C}$, with $\alpha$ being a root of $1+x+x^2$, does this system of equations have any exact solutions when $\lambda_4\neq 0$?
\begin{align}
-\lambda_1^2+\lambda_2\lambda_6&= \alpha^2\...
- 41
2
votes
1
answer
36
views
System of linear equations with a free variable.
I am new to linear algebra. The following matrix $A_n$ is an $(n-1) \times n$ matrix. I need to solve $A_nX = 0$, where $X = (x_1, \cdots, x_n)$.
\begin{equation}
A_n = \begin{bmatrix}
b_2 & a_2 &...
- 931
0
votes
1
answer
47
views
How to use Cramer's rule for $AX=0$?
For a general system of equation $AX=b$, Cramer's rule states to obtain $x_j$, we need to replace the $j$th column of $A$ with $b$, let us name this matrix as $D_j$ then calculate
$$
\dfrac{\det(D_j)}{...
- 931
4
votes
1
answer
87
views
What does the equation $x^2+y^2=r^2$ represent when $x, y, r$ are complex numbers?
I know this question is vague or maybe broad and subjective.
But, I am interested in studying the equation $x^2+y^2=r^2$ when $x,y,r$ are complex numbers.
What are a few directions that I can follow ...
- 73
0
votes
1
answer
36
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System of equations with sin and cos
So im currently trying to find the extremas of the function $f(x,y) = \cos(x+y) + \sin(x) + \sin(y)$
I've already computed the partial derivatives:
$$
f_x(x,y) = \cos(x) - \sin(x+y)\\
f_y(x,y) = \cos(...
- 101
0
votes
0
answers
31
views
Error in solving equations
I'm trying to solve for the constants in the following rational equation, which models position versus time data:
$$y = \frac{bt}{1+ct+dt^2}$$
The derivative or instantaneous velocity is given by:
$$\...
- 55
2
votes
1
answer
80
views
Solution of system of nonlinear equations with trigonometric terms
Issue:
I am trying to solve the following system of nonlinear equations for the unknown variables: $x$, $z$ and $\beta$. The remaining variables are known values.
$$a=u(s^2+(x\cos\beta\ )^2+(z\sin\...
- 21
1
vote
1
answer
128
views
Why homogeneity of an equation is preserved even when we change variables?
Consider the equations,
$$x^{2}+y^{2}+z^{2}-xt-t^{2}=0 \tag{1}$$
$$x^{2}+y^{2}+z^{2}+yt-2t^{2}=0 \tag{2}$$
Clearly, both equations are homogeneous. Solve for $t$ from the above equations. You will get ...
0
votes
1
answer
15
views
Indices without a given domain in sum notation.
I'm learning about normal modes right now and I have a question pertaining to the way the equations of motion are written vis-a-vis summation notation. The book defines the equations of motion of a ...
- 2,086
0
votes
1
answer
72
views
Setting up and solving an equation of torque given some data
A heavy hatch on a ship is made of a uniform plate of steel that measures 1.2 m X 1.2 m and has a mass of 400 kg. The hatch is hinged along one side;
it is horizontal when closed and opens upward. A ...
- 19
0
votes
0
answers
28
views
Cyclic Reduction of Linear Systems
I am stuck with a very complex problem related to cyclic reduction. I studied an example related to cyclic reduction of tridiagonal systems from the book Parallel scientific computing in C++ and MPI ...
- 111
1
vote
2
answers
115
views
Solve for $x$ and $y$ :
$$bx^3 = 10a^2bx + 3a^3y$$
$$ay^3 = 10ab^2y + 3b^3x$$
I tried putting $bx = p$ and $ay = q$ resulting into system of equations:
$$p^3 = 10a^2b^2p + 3a^2b^2q$$
$$q^3 = 10a^2b^2q + 3a^2b^2p$$
...
- 67
0
votes
2
answers
47
views
How to solve "linear multiset equations"?
I have a bunch of linear forms in the same number of variables, with the number of linear forms much larger than the number of variables. Say, they are $\ell_i(x_1,...,x_n)=l_{i1}x_1+...+l_{in}x_n$, ...
- 1,533
0
votes
0
answers
26
views
An entrepreneur sells for $100,000 and invests it in three accounts for one year. Linear system equations & simple interest
I need help with this exercise with constructing the system of equations.
An entrepreneur sells a portion of their business for $100000$ dollars and invests it in three accounts for one year. The ...
- 1,083
2
votes
2
answers
74
views
When does the equations have $1,2,3$ solutions?
There is given an equation,
$$ \frac{x^2-x+1}{x^2+x+1}=kx+1$$
When does this solution have one, two, three real solution(s) in $x$.
My Approach:
The above equation can be rewritten as,
$$\frac{-2x}{x^...
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votes
0
answers
34
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Confusion with pivot problem from Strang's Linear Algebra [P14 section 3]
This is the 14th problem in the 3rd section of the book: "Introduction to Linear Algebra" by Gilbert Strang.
The problem states: "Suppose Column 1 + Column 3 + Column 5 = 0 in a 4 by ...
0
votes
1
answer
61
views
Is there a general approach for solving a system of N cubic equations?
Is there a general efficient method/approach to solve a system of $N$ cubic equations in reals such that the number of variables $v$ are less than $N$.
I am aware of the simple approach for linear ...
- 41
-1
votes
0
answers
39
views
System of linear equations in 4 unknowns
I'm in first year of Information Systems Engineering, this problem is from Algebra and Analytic Geometry.
During the current year, 164 students entered UTN San Francisco to study Electromechanical (X),...
2
votes
6
answers
100
views
How many $(x,y)$ solutions does the system $\begin{cases}3^x+4^y=13 \\ \log_3x - \log_4y=1\end{cases}$ have?
How many $(x,y)$ solutions does the system $\begin{cases}3^x+4^y=13 \\ \log_3x - \log_4y=1\end{cases}$ have?
As I tried to solve this problem, I noticed that there is a single pair $(x,y)$ for which $...
- 833
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votes
0
answers
35
views
Vandermonde submatrix corresponding to $n$th roots of unity and efficiently solving linear system of equations
Let $n>k$, $n|q-1$. We have an $k*k$ linear system of equations $Ax=b$ over a finite field $\mathbb{F}_q$. Matrix $A$ is full rank and it is a submatrix of Vandermonde matrix $V$ corresponding to $...
- 41
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votes
0
answers
26
views
Geometric interpretation of an underdetermined system of equations
I am having a difficult time connecting different parts of a geometric interpretation of an underdetermined system of equations.
Given the matrix
$$A = \begin{bmatrix}
2 & 1 & 3\\
1 & 2 &...
- 123
2
votes
1
answer
58
views
Find solution of linear equation when knowing relation between columns
The exercise I have to solve is the following:
"Suppose $A$ is a $5\times 4$ matrix with columns $a_1, a_2, a_3, a_4$, and $a_1 + a_4 = a_2 + a_3$.
Which of the following vectors is (certainly) a ...
- 33
0
votes
1
answer
51
views
Evaluation of linear system conditioning
I am working in Matlab trying to solve a wider problem, that led to 2 linear systems
\begin{equation}
A_1 x = b_1^0
\end{equation}
\begin{equation}
A_2 x = b_2^0
\end{equation}
The matrices and ...
-1
votes
1
answer
34
views
Does there exists soutions to this system of trigonometric equations? [closed]
For two given integers $k,l\in\mathbb{N}$ and any real values $A,B,C,D$ satisfying $A^2+B^2+C^2+D^2=1$, does the following system of equations has a solution, on $x,y\in\mathbb{R}$?
$\cos(kx)\cos(ly)=...
- 11
0
votes
0
answers
38
views
Determinant of a binary matrix
So I've been trying a problem and I have reduced it to showing that the determinant of some binary matrix (the matrix filled with entries $0$ and $1$) is not equal to zero. I don't really know linear ...
- 520
1
vote
0
answers
40
views
Infinite linear system [duplicate]
I have to solve this system:
\begin{equation}
\begin{cases}
-R x_0 + w_1 x_1 + v_2 x_2 = 0 \\
R x_0 -(v_1 + w_1 + R) x_1 + w_2 x_2 + v_3 x_3 = 0\\
\vdots \\
R x_{i-1} -(v_i + w_i + R) x_i + w_{i+1} x_{...
- 21
0
votes
0
answers
19
views
General way to obtain Green’s function for simultaneous linear PDEs
Let’s say we have 2 unknown variables that are functions of 1D-space and time, $y(x,t)$ and $z(x,t)$.
Those two variables are in two simultaneous linear PDEs, let’s say
$$
\frac{\partial y}{\partial t}...
- 825
0
votes
0
answers
23
views
solving multi equation with different variable
I study Lagrange multiplier technique from 1 website. In the example, it is shown the steps and come out with these equations:
$2x + 𝜆_1 + 2x 𝜆_2 = 0$,
$8y + 𝜆_1 + 2y 𝜆_2 = 0 $
$x + y = 0$
$x^2 + ...
- 133
1
vote
1
answer
44
views
Linearity of system of differential equations?
I am learning how to solve differential equations (ordinary and partial)and why they are so important for physics.One thing I have noticed so far is that we know so little on the nature of the ...
- 21
0
votes
0
answers
22
views
How do we derive our range of values in plotting graphs of two systems of equation?
This is a topic that has to do with plotting graphs of simultaneously linear and quadratic to find x and y. In most questions or examples, we are usually given range of values as in (A) below while ...
- 1
0
votes
0
answers
35
views
A question about systems of nonlinear equations
Recently, I read a book on cluster algebra and come across a problem that could finally be reduced to a problem of solving a system of nonlinear equations.
The question is:
Give $b_{12},b_{13} , b_{...
- 737
1
vote
0
answers
66
views
Solution existence of general multivariable quadratic equations
Consider the variables $\mathbf{x}\in\mathbb{R}^n$ and the known coefficients $\mathbf{A}_i \in \mathbb{R}^{n\times n}, \mathbf{A}^T = \mathbf{A}, \mathbf{b}_i \in \mathbb{R}^n,$ and $c_i \in \mathbb{...
- 79