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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0
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1answer
22 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
26 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 ...
2
votes
1answer
57 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
23 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 $v$ indices;...
1
vote
0answers
25 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
38 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) $(y^2-1)x^2-...
1
vote
3answers
72 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 x\...
1
vote
1answer
40 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
103 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
65 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
15 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
37 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
10 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
27 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
36 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, $$y=-\frac{dx}{dt} \implies \frac{dy}{dt}=-\frac{d^2x}{dt^2}$$...
2
votes
1answer
35 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 variables,...
0
votes
1answer
66 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 ...
-1
votes
1answer
74 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
25 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 ...
0
votes
1answer
45 views

exponential equation system without log [duplicate]

How should I solve this equation system without using logarythms,using just a simple method? (E.g. turning it into a quadratic one using t) $$\left(\frac{3}{2}\right)^{x-y} - \left(\frac{2}{3}\right)^...
0
votes
1answer
28 views

Find a basis and the dimension of the solution space W of the following homogeneous system [closed]

Good morning, I need help with this problem. Find a basis and the dimension of the solution space $W $of the following homogeneous system $\begin{cases} x+2y-2z+2s-t=0\\ x+2y-z+3s-2t=0\\ 2x+4y-7z+s+...
3
votes
2answers
370 views

Solve 6 simultaneous equations for 8 variables puzzle

How to solve this puzzle? The image was sent to me with a caption in Chinese (解了一天了 帮帮忙吧… - googling leads to some solutions) and blank spaces where I have added letters. Separating each row and ...
-1
votes
1answer
40 views

Finding a and b from $a+b/3 = 1$ and $a/2+b/4=3/5$

I have two equations of which I need to solve for $a$ and $b$. $$ a+b/3=1\\ a/2+b/4=3/5 $$ Find $a$ and $b$.
4
votes
2answers
129 views

Prove that this system of linear equations generates $\left| \left( \begin{matrix} 1/2 \\ n \end{matrix} \right) \right|$ as a solution?

This infinite system of linear equations: $$ \begin{array}( 2x_1=1 \\ 3x_1+4x_2=2 \\ 4x_1+5x_2+6x_3=3 \\ \cdots \end{array} $$ In other words, this is particular case of a system: $$ \begin{array}( ...
0
votes
1answer
31 views

Problem of simplification

When trying to solve the equation $y^y = \frac{\ln^{y(1+c)}n}{n}$ , I've found the result $$y=\frac{-\ln n}{W(-\ln^{-c}n)}$$ where $c$ is a positive constant and $W$ is the Lambert function. The ...
0
votes
1answer
24 views

Using Cramer's Method

How do I use Cramer's method to solve the following system of equations ? 2x+2=10 2y=2 2-3y=6x I've solved for y using standard simultaneous equations but this didn't help
0
votes
2answers
32 views

Solving simultaneous equation without dividing by $x$

I have the following equations: $$3x^2 \dfrac{\partial x}{\partial u} + 3y^2 \dfrac{\partial y}{\partial u} = 1$$ $$y\dfrac{\partial x}{\partial u} + x\dfrac{\partial y}{\partial u} = 1$$ where I ...
-1
votes
1answer
17 views

Nonlinear autonomous system of differential equation [closed]

I have the following differential system : $x' = \frac {x^2}{(y-1)}$ $y' = x+1$ By elimination method things get ugly, so how could I solve it?
0
votes
0answers
15 views

Solving Non-Linear Simultaneous Equations

Equation 1: $$ L_{5}^{2} = (C_0 - C_2)^2 + (C_1 \cdot \sin(\theta_2) - C_3 \cdot \sin(\theta_1))^2 + (C_1 \cdot \cos(\theta_2) - C_3 \cos(\theta_1)))^2 $$ Equation 2: $$ L_{7}^{2} = (\lambda \cdot ...
2
votes
2answers
46 views

Do four points lie on the circumference of a single circle? Can I solve this with matrices?

I think I managed to figure out a way to determine whether three points lie in a single line via matrix determinants (but correct me if there's a problem): Where $y - mx - b = 0$, I plug each of the ...
0
votes
1answer
40 views

solution of system of polynomials

I have 3 equations as following: $$ \left\{ \begin{array}{c} (\Delta_{11}*y^2 + \Delta_{12}*y + \Delta_{13})x^2 + (\Delta_{21}*y^2 + \Delta_{22}*y + \Delta_{23})x + \Delta_{31}*y^2 + \Delta_{32}*y + \...
0
votes
1answer
17 views

Jacobian of a system of equations

I'm asked to compute the Jacobian of a system of equations $x_1^4+x_2^4-1=0$ $x_2-\sin(5x_1)=0$ $x_1-x_3^2=0$ What's the Jacobian of a system of equations? Do I perhaps need to infer the individual ...
0
votes
1answer
17 views

How to combine equations on xy plane with those on yz and zx plane?(diagram included)

I am considering a situation, where a cylinder is placed horizontally, like this (b(t) is the minimum distance between the circle and the ground) so, in the xy plane, I have the equation of a circle,...
0
votes
1answer
34 views

Method of undetermined coefficients for non-homogeneous linear system with two constant vectors

Suppose I have a system of non-homogeneous linear first order differential equations: $$ x'=A x+b_0+b_1t $$ Where $A$ is a $2\times2$ invertible matrix, $b_0$ and $b_1$ are: $$ b_0 = \begin{pmatrix}...
1
vote
0answers
65 views

Analytic solution for system of trigonometric equations

I have two equations as follows: $$ \left\{ \begin{array}{c} (\Delta_{11}\cos(\alpha) + \Delta_{12})\cos(\theta) + (\Delta_{21}\cos(\alpha) + \Delta_{22})\sin(\theta) = \Delta_{31}\sin(\alpha) + \...
4
votes
1answer
95 views

An equation over a finite field

Suppose $x,y,z,w \in \mathbb{F}_{q^2}$, where $q=p^k$ for some prime $p$. Consider the system of equations $$ \left\{ \begin{array}{l} xy + zw = 0; \\ xy^q + yx^q + zw^q + wz^q = 0. \end{array} \...
0
votes
0answers
34 views

How to represent a system of quadratic equations in matrix form

Suppose I have two quadratic equation like the following: $2x^2 - 3x + 2$ $x^2 + 5x + 6$ I want to find the minimum values of these equation with the constraint that: $-3 \lt x \lt 5$ How ...
0
votes
1answer
26 views

Find a system of recurrence relations

Find a system of recurrence relations for the number of $n$-digit binary sequences with $k$ adjacent pairs of $1$s and no adjacent pairs of $0$s. Any help on how to go about doing this would be ...
0
votes
0answers
14 views

Constructing a formula

I have two matrices A,B each item in A is tested against all items in B. If (a,b) matches then I have to calculate a weight for ...
2
votes
2answers
76 views

What can we do to solve the following equation with $6$ variables with some information provided?

Q) There are unique integers $a_2, a_3, a_4, a_5, a_6, a_7$ such that $$\frac{a_2}{2!}+\frac{a_3}{3!}+\frac{a_4}{4!}+\frac{a_5}{5!}+\frac{a_6}{6!}+\frac{a_7}{7!}=\frac 57$$,where $0\le a_i < i$. ...
1
vote
1answer
32 views

Solve this system using elimination for $x(t)$, $y(t)$

Here are my system of equations: $$x'+y'-x=5$$ $$x'+y'+y=1$$ I rearranged them like so: $$x=x'+y'-5$$ $$y=1-x'-y'$$ I took the derivative of $$x=x'+y'-5$$ and got $$x'=x''+y''\Rightarrow y'...
0
votes
4answers
35 views

How do I solve system if $2y−z=0$ and $2x+y−z=0$? [closed]

How do I solve system if $2y−z=0$ and $2x+y−z=0$?
0
votes
0answers
24 views

Basic linear system of equations clarification

I am trying to get a solid handle on linear systems of equations and I wanted to know if my following summary is correct. Linear systems of equations $Ax = b$ can either be square, overdetermined or ...
2
votes
1answer
114 views

Solving a system of algebraic and transcendental functions

I am attempting to solve a puzzle (LINK). As I have only taken up to multivariable calc, my college course knowledge hasn't helped. How do you solve a system of algebraic and transcendental functions? ...
0
votes
1answer
30 views

How to use elimination to solve a system of equations with 3 variables

I asked this question before but with a linear algebra angle, however, I need to solve it using the elimination method. I have been able to solve for $y$, which is $y=\frac{-y''}{2}+\frac{3y'}{2}$, ...
0
votes
0answers
22 views

Analytical Case Differentiation

is there a analytical way for case differentiation? In my case a MonteCarlo Simulation calculates a system of equations. Parameters can randomly change so that the underlied mathematical condition ...