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.

Filter by
Sorted by
Tagged with
1 vote
1 answer
12 views

finding the critical points of a system of equations

I have a question from a past paper where I am asked to find the critical point(s) in the second ($x<0$ & $y>0$) and third quadrant ($x<0$ & $y<0$) of the following system: $$ X=x^...
-1 votes
0 answers
13 views

Correctness of direct numerical solution of ill-conditioned linear system

To what extent can you put trust in a numerical solution obtained by direct solver for an ill-conditioned linear system? In other words, how can you check the solution? Dropping it into the system ...
3 votes
2 answers
87 views

Geometry - Solve a system using Wolfram Alpha

I found this problem on the web, and it seemed like an interesting exercise to me. So I am trying to solve this as a system of equations using Wolfram Alpha. That means e.g. I am trying to find the ...
  • 11.7k
0 votes
4 answers
421 views

How to solve this quadratic system of equations?

The equations are: $$ \begin{aligned} b + d &= c^2 - 6 \\ b - d &= -\frac{1}{c} \\ b d &= 6 \end{aligned} $$ and I want integer solutions for this. I tried using various methods such as ...
3 votes
1 answer
44 views

A system of modular equations

Is there a way to solve for $a,b,c,d$ in the following system? $$ \begin{aligned} \begin{cases} ac & \equiv P \\ ad & \equiv Q \\ bc & \equiv R \\ bd & \equiv S \end{cases} \pmod{M} \...
  • 1,196
0 votes
0 answers
13 views

Simultaneous equations in non-Euclidean space?

As I recall, one visualizes equations as lines or planes in Euclidean space and the solutions are intersections among these lines, planes or higher-dimensional equivalents. Is there some use to ...
  • 185
0 votes
0 answers
21 views

Set of 5 equations for 6 positive integer unknowns

The setup is that we have a total of 9 persons scattered in 6 groups. Each persons having a unknown number of pens and being part of two different groups. The repartition of the groups is like so : $$\...
2 votes
1 answer
111 views

A problem on sums of powers

For $n \ge 3,$ if $$\sum_{k=1}^{n-1}x_k^m = y_m = (m+1)n-2, \quad \forall 1 \leqslant m \leqslant n - 1$$ then $$\sum_{k=1}^{n-1}x_k^n = y_n = n^2+n-2.$$ Essentially, $y_1,y_2,y_3 \ldots y_n$ are in ...
1 vote
1 answer
39 views

How to simplify to remove $\log(n)$ term?

How can I simplify the following equation to get rid of the $n$ term on both sides and get a value for the constant $d$: $$\frac{2}{4}\log\left(\frac{n}{4}\right) = d\log(n)$$ Is it possible to get ...
0 votes
0 answers
15 views

If $A \in \Bbb C^{n\times n}$, Show that $L(A, b) \neq \emptyset$ iff $b \in L(A^H, 0)^\perp$

If I put the problem into other words, then $Ax = b$ has a solution iff b is in the orthogonal complement of the kernel of $A^*$ i.e. the complex transpose of A. I know how to do these problems when ...
  • 65
-4 votes
0 answers
116 views

If $ {\bf A} {\bf x} = {\bf B} {\bf x}$ for all ${\bf x} \in \Bbb R^n$, is ${\bf A} = {\bf B}$?

If $ {\bf A} {\bf x} = {\bf B} {\bf x}$, where $\bf x$ is a vector and $\bf A$ and $\bf B$ are matrices, for all ${\bf x} \in \Bbb R^n$, then ${\bf A} = {\bf B}$. For this theorem, is it not possible ...
-3 votes
0 answers
17 views

Gaussian Elimination & Substitution Question [closed]

I am stuck on the following question. I can get 3 simulation equations from (a) but am unsure how to progress from there.
  • 25
1 vote
1 answer
97 views

How do we know if a system of equations has multiple solutions and how to solve it?

For example, my equations are $\lambda_1=\sqrt{\dfrac{\alpha^2}{2}+\sqrt{k^4+\dfrac{\alpha^4}{4}}}$ $\lambda_2=\sqrt{-\dfrac{\alpha^2}{2}+\sqrt{k^4+\dfrac{\alpha^4}{4}}}$ $\lambda_1\tan\!\big(\!\...
0 votes
0 answers
13 views

Does every augmented matrix have a unique set of free variables?

For example: $x + y = 1$ $z + y = 1$ Looking at this I'd think that $y$ is the free variable. But we could also do: $y = 1 - x$ $z = 1 - y = x$ Wouldn't $x$ be the free variable here? Doesn't this ...
0 votes
1 answer
39 views

Creating and Solving Equation from Word Problem

I have this word problem that I cannot figure out how to put into an equation: Suppose that we agree to pay you $9$ cents for every math problem you solved correctly, and fine you $5$ cents for every ...
1 vote
0 answers
21 views

Transforming a of system of differential equations of variable coefficients into system of constant coefficients

I want to change the following system into a system of constant coefficients. $x_1'=(1+\cos2t)x_1+(1-2\sin2t)x_2$, $x_2'=-(1+2\sin2t)x_1+(1-2\sin2t)x_2$ In the question itself, they have given a hint ...
0 votes
1 answer
84 views

Resolve f(x+1)=f(x)+100 [closed]

I have the following data: ...
0 votes
2 answers
30 views

Unique pair of positive real numbers satisfying $x^4 -6x^2y^2 + y^4 = 8$ and $x^3y - xy^3 = 2\sqrt{3}$

(Mandelbrot) There is a unique pair of positive real numbers satisfying the equations \begin{equation} x^4 - 6x^2y^2 + y^4 = 8 \hspace{1em} \text{and} \hspace{1em} x^3y - xy^3 = 2\sqrt{3}\text{.} \end{...
-2 votes
1 answer
57 views

How to solve system of equations $cosy+x=ycosx-sinx\sqrt{1-y^2}$,$cos(y+x)=y$? [closed]

$$ \cos y+x=y \cos x-\sin x\sqrt{1-y^2}$$ $$\cos(y+x)=y$$ $$\implies x=0,\ y=D \ (\text{Dottie Number})$$ Plot
-1 votes
0 answers
23 views

Parametric Vector Form vs Notation [closed]

I come across a lot of exercises asking to find the general solution of a homogenous system of equations but the type of answer confuse me often between Parametric Vector Form and Parametric Vector ...
0 votes
0 answers
22 views

Solve system of polynomial equations based on distance?

I am trying to find the solution for this polynomial system. For some context it represents a game where every player has to choose a location in the x-line, given that it is best to be closer to 0, ...
  • 205
0 votes
1 answer
56 views

Solve for rational coefficients

Let $n \geq 0$ be and even integer. I have $2n + 5$ data points $(x_i,y_i)$ for $i = 1,\ldots,2n+5$. I wish to find parameters $r, a_{n+1},\ldots, a_0, b_{n-1}, \ldots, b_0, c, d$ (I will call these ...
  • 21
2 votes
1 answer
72 views

Solving $x=\frac{\alpha y}{\alpha y+\beta z}$, $y=\frac{\gamma x}{\gamma x+\delta(1-z)}$, $z=\frac{\epsilon(1-x)}{\epsilon(1-x)+\zeta(1-y)}$

Given this set of three rational equations: $$ x = \frac{\alpha\cdot y}{\alpha\cdot y + \beta\cdot z} $$ $$ y = \frac{\gamma\cdot x}{\gamma\cdot x + \delta\cdot (1-z)} $$ $$ z = \frac{\epsilon\cdot (1-...
3 votes
1 answer
43 views

How to find $L$ if $L=\frac{c}{(1-L)^a}$

How to find $L$ if $L=\frac{c}{(1-L)^a}$ I was trying to apply log but $\ln L +a\ln (1-L)=\ln c$. How can continued please? Thank you
0 votes
0 answers
25 views

Find all vectors y such that Ax = y is inconsistent

I am given a matrix A, but that is it. The question is to find all vectors y such that Ax = y is inconsistent, where x is a vector as well (no values are given). My first thought is that there is an ...
  • 17
0 votes
0 answers
23 views

Finding integral points in a rational lattice

Here is a reduced Gröbner basis of a zero-dimensional ideal $I$ in Singular format, one of whose solutions $(q,a,b)$ provides the key numbers in the solution to the Six Disks Problem: ...
  • 93.9k
1 vote
2 answers
74 views

Proof of linear independence of$~\boldsymbol{{x_1\over\Vert\boldsymbol x_1\Vert},x_2} ~$where$~\boldsymbol{x_1,x_2}~$are linearly independent.

$$ \boldsymbol{x_1},~\ldots,\boldsymbol x_s:=n~\text{dimensional real vectors which are linearly independent} $$ And it is obvious that there is no such$~\boldsymbol x_i=\boldsymbol 0~\text{for}~i\in\...
3 votes
2 answers
242 views

Solving an infinite system of equations for the coefficients of a power series for $f(x+1)=\exp(f(x))$

Consider the sequence of formulae, $a_n$, such that $a_0=1$ and for all $n>0$, we have $$a_n = \sum_{k=1}^nT(n,k)c_k a_{n-k},$$ where $$T(n,k)=\frac{(n-1)!k}{(n-k)!},$$ $c_0=0$, and $c_k$ for $k>...
  • 3,744
1 vote
1 answer
21 views

4 simultaneous equations, 3 unknowns, How do I know I've found all the solutions?

$$4x-y-z=21$$ $$2x+4y+z=69$$ $$8x+y-z=81$$ $$-4x+7y+3z=57$$ Solutions are $y=30-2x$ and $z=6x-51$ according to wolfram alpha and it's quite simple to get to these solutions and then check that they ...
0 votes
1 answer
28 views

Predict stable or cyclic population variation in dynamical system

Does the Lotka-Volterra model predict stable or cyclic population variation? What determines the amplitude of the cycles predicted by the Lotka-Volterra model? The Lotka–Volterra equations, also ...
  • 537
4 votes
1 answer
103 views

Determine if weighted graph can be physically constructed, treating weight as Euclidean distance (ie check if subset of distances is self-consistent)

Suppose we want to position some points in space, given that we know at least some of the distances between them. How can we determine if this is possible? And if it is possible, can we determine the ...
  • 1,295
0 votes
0 answers
53 views

How to solve ODEs with a parameter that changes over time.

I have the following coupled ODEs: $$ \begin{cases} \frac{dA}{dt} & = k_{1}\text{activity} - k_{2}A\\ \frac{dB}{dt} & = k_{3}A - k_{4}B \end{cases} $$ with $k_1 = 1.0, k_2 = 0.1, k_3 = 0.1, ...
0 votes
1 answer
40 views

Tedious four variable equations

I am trying to calculate something very tedious. I got this after calculating a Lagrangian equation with three multipliers. I need to write each of $k_x, k_y, l_x, l_y$ in terms of $\overline{x}$ and $...
  • 72
4 votes
1 answer
64 views

Covering unit square with discs; systems of degree-2 polynomials

Given a unit square and $n$ identical circles/discs, what is the smallest radius $r_n$ for which the circles can fully cover the square? For $n=4$, the proven minimal solution is $r_4 = 1/(2\sqrt2) \...
  • 2,101
-1 votes
1 answer
27 views

Linear Algebra: How do I find the total trips for each area given the 24 hours limit. [See problem below]

You run a delivery company, delivering in three different areas of Manhattan, A, B, and C. On average, a trip to the area takes 4 hours, 5 gallons of fuel and you deliver 3 tons of goods. A trip to ...
0 votes
1 answer
26 views

All numeric solutions of system of nonlinear equations [closed]

I want solve 2D system nonlinear equations. First method is multidimensional Newton (derivatives can be computed). It is fast and general. But is not always convergent, especially if I don't know how ...
  • 129
0 votes
1 answer
20 views

Necessary and sufficient conditions for an existence of an orthogonal matrix$P~$ s.t. $~P^{-1}AP~$is diagonal, using$~a~$which is one of entries of$A$

This problem is quoted from the $3$rd year transfer exam of math major in the university. $$\begin{align} a:=\text{real number}\\ A:= \begin{pmatrix} 0&a&2\\ 1&0&2\\ 2&2&3 \end{...
0 votes
0 answers
30 views

Solutions to systems of quadratic multivariate polynomials with diagonal quadratic forms

I am looking for an analytical solution to a system of quadratic equations of the form: $$ \mathbf{x}^T \mathbf{A}_i \mathbf{x} + \mathbf{b}_i^T\mathbf{x} + c_i=0 ~~~~~~~~ i = 1,..,n $$ where $\...
  • 19
1 vote
2 answers
54 views

Solve system for elements of a matrix

I have a system of $n$ equations which follows a particular pattern as follows (showing the case $n=3$): $$\phi = a_1 + \psi_2 a_2 + \psi_3 a_3 \\ \phi = \psi_1 a_1 + a_2 + \psi_3 a_3\\ \phi = \psi_1 ...
  • 137
1 vote
0 answers
33 views

Calculate Specific Rotation Matrix to align Vector A to Vector B in 3d?

I have a question very similar to this post (that I found quite helpful), but slightly different. Calculate Rotation Matrix to align Vector A to Vector B in 3d? I would like to accomplish the same ...
0 votes
0 answers
26 views

Solving a system of PDEs with an ODE

I want to solve the following system of equations which consists two PDEs and one ODE: \begin{align} \rho_t+v\rho_x &= 0; \newline Y_t+vY_x &= 0 ;\newline v_t &= -\frac{1}{(\...
1 vote
1 answer
56 views

Solve the nonlinear system with three equations and three variables $x,y,\lambda$.

$\begin{cases} \dfrac{x}{\sqrt{x^2+(y-y_A)^2}} + \dfrac{x-x_B}{\sqrt{(x-x_B)^2+(y-y_B)^2}} = 2x\lambda \\\\ \dfrac{y - y_A}{\sqrt{x^2+(y-y_A)^2}} + \dfrac{y-y_B}{\sqrt{(x-x_B)^2+(y-y_B)^2}} = 2y\...
user avatar
-1 votes
0 answers
23 views

Solving system of equations that have an "or" relationship

I have the following problem to solve: Either John and I can do the same number of one-legged squats, OR John can do infinitely more than I can How many squats can I do? I attempted to solve it like ...
0 votes
1 answer
95 views

Evaluation of all solutions such that $~ y(x)>0 ~$ for $~ x>0 ~$where $~y(x)~$ is a general solution of $2$nd order linear nonhomogeneous DE

The essential problem statement is shown far below this post with bold italic font. $$ \text{Evalution of solution}~ y(x) ~ \text{of}~ y''+\sqrt{5}y'-y+2=0 $$ My works $$\begin{align} y''+\sqrt{5}...
0 votes
1 answer
33 views

When is the $B$ defined through $\forall i:a_i=v_i\times B$ existent and unique?

Of course the motivation is to understand what data is needed to define the (electric and) magnetic field through the lorentz force. So suppose we are given $a_1,\ldots,a_n,v_1,\ldots,v_n\in\mathbb R^...
  • 2,197
0 votes
0 answers
16 views

Prove the existence of solution for a special linear system of equations

Assume there is a infinitive sequence $a_0,a_1,a_2,\cdots$ belonging to a field $F$.For every nonnegative integer $s,m$,define $$ A_{s,m}=\begin{bmatrix} a_s & a_{s+1} & \cdots & a_{...
2 votes
0 answers
56 views

Is there a general method for finding real solutions to this class of systems of equations?

Fix a positive integer $n$. Let $\mathbf f=(f_1,\ldots,f_n)$ such that for each positive integer $i\le n$: the function $f_i:\Bbb R^{n-1}\to\Bbb R$ is a polynomial function with real [real algebraic?]...
  • 4,102
0 votes
0 answers
21 views

I want to solve system of N first order non-linear differential equations using Runge-Kutta methods. Please help me for regarding the same?

I know how to solve the system of linear ODEs using Runge Kutta, but very doubtful about the non-linear ones. Can I use the same method as the linear one in solving the non-linear system of ODEs? I am ...
3 votes
2 answers
92 views

What is wrong in this solution of $\sec x + \csc x = 2 \sqrt{2}$

Find number of solutions in the interval $[0,2\pi]$ of the equation - $$\csc x + \sec x = 2 \sqrt{2}.$$ $⇒ \dfrac{1}{\sin x}+ \dfrac{1}{\cos x }= 2 \sqrt2$ $⇒ \dfrac{\sin x +\cos x}{\sin x \cos x} = ...
  • 180
1 vote
0 answers
24 views

Prove for Existence and Uniqueness of Linear Equation

Is that a good prove? Am I missing something? Consider a system with n >= 1 linear equations: $$ \left\{ \begin{array}{c} a_1 * x = B_1 \\ ... \\ a_2 * x = B_2 \\ \end{array} \right. $$ in which $...
  • 135

1
2 3 4 5
155