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
29 views

What values of 'a' and 'b' would create a unique, no, and infinite solution(s)?

The following problem has really troubled me: I have row reduced it, so now it looks more like this: How do I really figure out what values of a and b would create infinitely many solutions, no ...
0
votes
1answer
23 views

System of equations premutations

When I permutate a,b,c,d,e to the left the value on the right side changes by 1. Can I make some use of this information to solve the following system of equations? I don't really know have to solve ...
0
votes
2answers
66 views

System of equations symmetric

How do I solve the following system of equation? $$ xyz = x+y+z $$ $$ xyt = x+y+t $$ $$ xzt = x+z+t $$ $$ yzt=y+z+t $$ I have no idea how to do.
0
votes
1answer
33 views

Systems of ODEs

I want to solve a system of ODEs of the following type: $$\large\frac{d\phi_{i}}{dt} = {\mu_{i}}^2\phi_{i} + \sum_{j=1}^{N}a_{ij}\phi_{j}$$ There were IMSL/Visual Numerics routines such as DMOLCH, ...
4
votes
2answers
164 views

How can I solve this hard system of equations?

Solve the system below \begin{align} &\sqrt {3x} \left( 1+\frac {1}{x+y} \right) =2\\ &\sqrt {7y} \left( 1-\frac{1}{x+y} \right) =4\sqrt{2} \end{align} Frankly I am disappointed, ...
1
vote
2answers
39 views

Is there an adjective to describe systems of equations which is neither underdetermined nor overdetermined?

What might I call a system of equations in which the number of equations equals the number of free variables? In other words, if a system of equations is neither underdetermined nor overdetermined, ...
0
votes
1answer
24 views

Calculate a in dependence of b so that equation system is solveable?

Given is following equation system: $\begin{pmatrix} 2 & 1 & 1 \\ 3 & 2 & 3 \\ 4 & 3 & 5 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} =\begin{pmatrix} a \\ b \\ 1 ...
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votes
2answers
32 views

Determine kernel of matrix

The following matrix is given: $A=\begin{pmatrix} 2 & 1 & -2 \\ 0 & 1 & 0 \\ 0 & 3 & 4 \end{pmatrix} $ a) Determine kernel of matrix A I did this, but I always end up with ...
0
votes
2answers
27 views

System of linear equations with parameter - strange result, does this make sense

I have a system of linear equations $Ax = b$ where $A \in \mathbb R^{3\times 3}$, and $x,b = \in \mathbb R^{3 \times 1}$. $A$ has some parameter $\alpha$ in its entries. I was asked to find for ...
0
votes
0answers
33 views

Get content of transformation matrix from transformed vectors

In the following example: $$ \begin{pmatrix} X\\ Y\\ \end{pmatrix} = \begin{pmatrix} \cos\alpha & 1\\ 0 & \sin\beta\\ \end{pmatrix} \begin{pmatrix} A\\ B\\ \end{pmatrix} $$ $X$, $Y$, $A$ and $...
0
votes
1answer
35 views

Solutions of a system of polynomial equations

I am trying to find the critical points of some functions such as $$f(x, y) = x^4 − x^2y^2 + y^3 − 18x^2 + 3y^2$$ I calculate the gradient, and then find a system of polynomial equations: $$\...
0
votes
1answer
37 views

Solutions to a linear system of equations.

I want to find the solution of this system when the parameter $a \in R$ varies. \begin{cases} (a+2)x_2 + x_4 = 1 \\ -x_1 +x_3 = a+1 \\ (a+1)x_1 + 2x_2 -x_3 = 0 \\ x_1 -2x_2 -(a+1)x_3 = -2 \end{cases}...
1
vote
0answers
51 views

Finiteness of solutions to system of polynomial equations $P(x)P(y)=1$ & $Q(x)Q(y)=1$

Can that finiteness be proved for polynomials $P^n\neq\pm Q^m,\quad n,m>0\;$ by known methods? For univariate polynomials $A,B,X,Y$ $$AX+BY\equiv0$$ iff $$A\equiv\frac{HY}{(X,Y)},B\equiv-\...
0
votes
1answer
30 views

Preconditioner operator

hope you can help me. I have learned that a preconditioner is a matrix $P$ such that when it is applied to a system $A \mathbf{x} = \mathbf{b}$, the spectral properties of the matrix $P^{-1} A$ are ...
1
vote
3answers
36 views

simultaneous equations help needed

I think of two numbers, x and y. When I add them together I get 5 and when I find the difference I get 13. What numbers did I think of? I need to know how to write this down in simultaneous equation ...
0
votes
1answer
33 views

System of linear congruence when not relatively prime

I am new to Abstract Algebra and understand how to solve when the mods are relatively prime, but I am struggling when they aren't relatively prime. I have a system of of linear congruences that I ...
3
votes
1answer
65 views

number of real triplets $(x,y,z)$ in system of equations

Total number of real triplets of $(x,y,z)$ in $x^3+y^3+z^3=x+y+z$ and $x^2+y^2+z^2=xyz$ $\bf{My\; Try::}$ Let $x+y+z=a$ and $xy+yz+zx=b\;,$ Then $(x+y+z)^2-2(xy+yz+zx)=xyz\implies a^2-2b=xyz$ And ...
0
votes
0answers
23 views

Solve system of polynomials

I have four polynomials with four unknowns $x_{1},x_{2},y_{1}$ and $y_{2}$ as following $$ \left\{ \begin{array}{c} m_{1} + m_{2}x_{1} + (m_{5} + m_{6}x_{2})y_{1} + (m_{3}+m_{4}x_{1})y_{2} ==0 \\ n_{...
0
votes
0answers
34 views

Hadamard product and linear systems

Given two matrices $A, B \in \mathbb{R}^{n \times m}$, where $A = \{a_{i,j}\}$ and $B = \{b_{i,j}\}$, the Hadamard product (or point-wise product) is a matrix $$C = A \circ B$$ such that $C = \{c_{i,j}...
2
votes
1answer
44 views

System of linear equations - geometrical representation of solution

So, if we're given the system: $$x+ay=1$$ $$-ay+z=a-1$$ $$2z=a$$ where $a\neq 0$, we can write it's solutions as: $$x=\frac{a}{2}$$ $$y=-\frac{1}{2}+\frac{1}{a}$$ $$z=\frac{a}{2}$$ or: $$(x,y,z)=\bigg\...
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0answers
18 views

Existence of solution for nonlinear (algebraic) equations.

Let $f_1(x_1,\cdots,x_n)=0$, $\vdots$ $f_n(x_1,\cdots,x_n)=0,$ be a nonlinear equation. Is there a condition on $f_1,\cdots,f_n$ under which this equation has a solution? Thanks for your help.
1
vote
1answer
70 views

First integrals for solving system of ODEs

Assume a problem $$ \begin{cases} \frac{\mathrm{dx}}{\mathrm{dt}} = \frac{y}{x-y}, \\[2ex] \frac{\mathrm{dy}}{\mathrm{dt}} = \frac{x}{x-y}. \end{cases}$$ Additionally, $x = x(t)$ and $y=y(t)$. ...
2
votes
2answers
51 views

Using augmented matrices to find a number

There's this system of equations $$(8 − a)x_1 + 2x_2 + 3x_3 + ax_4 = 2$$ $$x_1 + (9 − a)x_2 + 4x_3 + ax_4 = 1$$ $$x_1 + 2x_2 + (10 − a)x_3 + ax_4 = 2$$ $$x_1 + 2x_2 + 3x_3 + ax_4 = 2$$ Now I have ...
1
vote
1answer
30 views

Showing that a system of equation is inconsistent

When working out this system of equations, I've found that there are no solutions. Is there any way from the start you can Identify that this system of equations has no solutions? $2x + y − z + u = ...
0
votes
0answers
51 views

System of equations for semi-unitary matrix

I have a semi unitary matrix $A_{i,j}$ with $1 \leq j \leq N$, $1 \leq i \leq M$ and $M\geq N$, i.e. $A^\dagger A = I$. I now have a set $N$ equations for the squared entries of each row: $$\sum_j |...
1
vote
2answers
35 views

A system of polynomial equations of degree $2$ in two variables

I need to find an explicit solution of this system of polynomial equations of degree $2$ in two variables $x,\,y$: $$\begin{cases} p_1x^2+q_1y^2+r_1xy+s_1x+t_1y+u_1=0\\ p_2x^2+q_2y^2+r_2xy+s_2x+t_2y+...
0
votes
2answers
30 views

How to numerically find zeros of a system of first-order differential equations (Airy function)?

To numerically approximate the Airy function y = Ai(x) which satisfies the equation $$ y'' - xy = 0 $$ I converted this second-order diff. eq. into a pair of first order diff eq. and solved them using ...
2
votes
2answers
54 views

Systems of equations with unknown constant

How do I solve this system? It says I must row reduce it to solve it (depending on parameter $a$). $(8−a)x_1 + 2x_2 + 3x_3 + ax_4 = 2$ $x_1 + (9−a)x_2 + 4x_3 + ax_4 = 1$ $x_1 + 2x_2 + (10−a)x_3 + ...
1
vote
1answer
40 views

Solve for two variable in terms of other [closed]

How do I solve for $\alpha$ and $\beta$ in terms of $\theta$ using the equations $$a^2 \cos^2\theta \:+\:b^2 \sin^2\theta \:=\:a^2\cos^2\alpha $$ and $$b^2 \cos^2\theta \:+\:a^2 \sin^2\theta \:=\:a^2\...
2
votes
0answers
36 views

Derivative solution of $\frac{(x\cos\,\theta-y\sin\,\theta)^2}{a^2}+\frac{(x\sin\,\theta+y\cos\,\theta)^2}{b^2}=1$

The equation $$\frac{(x\cos\,\theta-y\sin\,\theta)^2}{a^2}+\frac{(x\sin\,\theta+y\cos\,\theta)^2}{b^2}=1 \ \ \ \ \ \ \ \ (*)$$ has the following expression for the derivative: $$\frac{\mathrm dy}{\...
0
votes
3answers
31 views

Algebraic system of equations problem

Solve the follow system of equations: $$x+y+z=5$$ $$\frac{1}{xy}+\frac{1}{yz}+\frac{1}{xz}=5$$ $$x^3+y^3+z^3=53$$ Thanks for any help.
2
votes
1answer
19 views

Chinese Remainder Theorem When GCD is not 1

I've got this system of equations that I'm trying to solve. I'm pretty sure I have to use the CRT, but to my understanding, it can only be used when GCD of all the mods is 1. I don't want an answer ...
1
vote
2answers
27 views

normalization of constraints $ 0 \leq x \leq 1 $ in Lagrangian KKT

With Lagrangian we have an objective function and a set of equality constraints of form $ g_{i}(x_{j}) = 0 $ . With KKT we can have another set of inequality constraints of the form $ h_{i}(x_{j}) \...
0
votes
1answer
20 views

System of congruences of 2 unknowns

Given constants $A, B, C, D$, and unknowns $x, m$, how would I go about solving a system such as this: $$A\equiv x B\mod m$$ $$C\equiv x D\mod m$$ I'm certain there is a very simple equivalent form of ...
3
votes
4answers
96 views

I'm stuck in a logarithm question: $4^{y+3x} = 64$ and $\log_x(x+12)- 3 \log_x4= -1$

If $4^{y+3x} = 64$ and $\log_x(x+12)- 3 \log_x4= -1$ so $x + 2y= ?$ I've tried this far, and I'm stuck $$\begin{align}4^{y+3x}&= 64 \\ 4^{y+3x} &= 4^3 \\ y+3x &= 3 \end{align}$$ $$\begin{...
1
vote
1answer
95 views

Solve the system of equations $x^y=y^x$

Solve the system of equations $$ x^y=y^x \\ a^x=b^y $$ I could not solve this despite many tries
0
votes
2answers
34 views

number of solutions of system of two equations, two unknowns (Matrix)

How can we find that when a system of two equations, two unknowns has Infinite Solutions. I want a solution with matrix. I know this method (which is not with matrix): $ax + by = c$ $a'x+ b'y = c'$ ...
0
votes
0answers
48 views

Solution to System of Complicated Differential Equations

I'm looking for a solution to this set of complicated differential equations: $$\begin{align} \dfrac{dθ}{ds} & = \dfrac{\cos θ}{r} − z\\ \dfrac{dz}{ds}& = − \cos θ \\ \dfrac{dr}{ds} &= \...
1
vote
1answer
19 views

How to determine if an object in space is pointing at (oriented toward) another object?

QUESTION: You know the position of two objects in space (one also has an orientation). How do you determine when the object is pointing/oriented at the other object? Hopefully this question makes ...
-2
votes
2answers
61 views

Simple but hard 2 by 2 system in $x$ and $y$ [duplicate]

Is there a systematic way of solving this system, analytically? $$\begin{cases} x \ + \ y^2=11\\ x^2+y\ \ =\ 7\\ \end{cases} $$ I mean, other than brute-force.
0
votes
0answers
29 views

Probabalistic solve of system of equations

I'm engineer, not mathematician, so excuse me for wrong terminology, but I hope you'll understand the problem. Example situation: I have N electronic components. Each of them has reactance and ...
3
votes
3answers
46 views

Maximize system of linear equations

Suppose you have the system $$ \begin{bmatrix} 4 & 3\\ 1 & 7\\ 5 & 9\\ 2 & 4\\ \end{bmatrix} \begin{bmatrix} x\\y\end{bmatrix}=\begin{bmatrix}b_1\\b_2\\b_3\\b_4\end{bmatrix} $$ How ...
0
votes
0answers
26 views

Solve the EDO $p'={\alpha}p^a+{\beta}p^b,\quad t>0,$

Fix $\alpha , \beta \in (0,\infty)$ . Use Osgood's criterion to show that the equation $$p'={\alpha}p^a+{\beta}p^b,\quad t>0,$$ has at most one nonnegative solution if $a,b \ge 1$. Also, prove ...
0
votes
1answer
18 views

Value of $a$ if system of equation is consistent.

If the following equations are consistent and have more than one solution, what is the value of $a$? Given $u+v=-(av+1)$ $u+2v=-a(v-1)$ $3u+8v=a+2$ I was thinking that system of equation is ...
0
votes
0answers
43 views

Linear system equations

I need to get, preferably by a numerical method, a solutions of: $$\left\{\begin{array}{lll} 2\sum_{i=1}^n b_{ik}x_i+x_{n+1}=0&\text{for}& k=1,2\ldots,n\\ \sum_{i=1}^n x_i=0 \end{array}\right.$...
0
votes
0answers
34 views

Solving system of nonlinear equations via iteration

I will give an example to illustrate the question: Assume I have the system: $$ xy + x + y = 7\\ x^2 + y^3 = 9 $$ and I want to solve for $x$ and $y$. It is a fairly common approach to rearrange ...
1
vote
2answers
38 views

Decouple a system of two second order differential equations

I have a system of second-order differential equations that I want to decouple. they are, $\ddot{x} = \frac{\omega_1^2}{2} x + \omega_2 \dot{y}$ and $\ddot{y} = \frac{\omega_1^2}{2} y - \omega_2 \...
1
vote
3answers
38 views

Process for solving this system of equations

I have this system of equations for which I'd like to solve for $x$,$y$, and $r$ where $a$,$b$, and $t$ are constants: 1: $0 = (x-a)^2 + (y - b)^2 - t^2$ 2: $y = \dfrac{bx-rx+ar}{a}$ 3: $r = \dfrac{...
0
votes
1answer
37 views

How to solve simultaneous inequalities (reasked)? [duplicate]

I am doing multivariable calculus, and specifically double integrals. I am facing difficulties finding the domain of the integal, however i am given the following equations: $$1≤2x+y≤2$$ $$0≤x−2y≤1$$ ...
-2
votes
1answer
45 views

Find an energy functional for the nonlinear viscous oscillator $x' = v$, $v' =-b(v)v-k(x)x$, $t>0$ [closed]

Consider the nonlinear viscous oscillator $$\begin{cases} x' = v\\ v' =-b(v)v-k(x)x,\quad t>0, \\ \end{cases}$$ where $(x,v)$ is the position and velocity of the oscillator. Here $b : \mathbb R\...