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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.

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Solving a System of Equations with Zero Determinant Matrix

I'm trying to learn FEA and I'm going through the first example problem from the "Practical Stress Analysis with Finite Elements" book by Bryan Mac Donald. Here is the problem: Matrix ...
Dean Justin Nunez's user avatar
1 vote
2 answers
39 views

Find all triplets $(a, b, c)$ of integers for which the following holds: $a^2 = bc + 1$ and $b^2 = ca + 1$. [duplicate]

Find all triplets $(a, b, c)$ of integers for which the following holds: $a^2 = bc + 1$ and $b^2 = ca + 1$. Attempt: First, I subtracted the two equations and obtained $(a-b)(a+b) = c(a-b)(-1)$. Now, ...
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-1 votes
0 answers
25 views

Finding the equation of a sine wave using two points and a tangent [closed]

I am trying to find a sine wave of form y=asin(b(x-c)) (a=/=0, b=/=0) that intersects the point (45, -25) with a slope of 1.55, and intersects the x-axis at x=52. using this information I made 3 ...
mmmm's user avatar
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1 vote
3 answers
135 views

Solve for real $x,y,z$ : $x^2 + xy + y^2 = a$, $y^2 + yz + z^2 = b$, $z^2 + xz + x^2 = c$ .

Solve for complex $x,y,z$ : $x^2 + xy + y^2 = a$, $y^2 + yz + z^2 = b$, $z^2 + xz + x^2 = c$ where $a,b,c \in R$ such that $a,b,c \ge 0$ . We've : $x^2 + xy + y^2 = a \dots(1)$ $y^2 + yz + z^2 = b \...
Ash_Blanc's user avatar
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-1 votes
1 answer
93 views

Example of functions $f$ and $g$ where $f\circ g(x) = x^2$ and $g \circ f(x) = x^3$ for range $(1, \infty)$ [closed]

I'm just looking for examples of real functions $f$ and $g$ where $f \circ g(x) = x^2$ and $g \circ f(x) = x^3$ and the domains and codomains of $f$ and $g$ is $(1, \infty)$. No complex functions but ...
Coach Jonathan Ramachandran's user avatar
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1 answer
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A man buys a dozen pieces of fruit...Questions about finding which solutions work [duplicate]

From Dudley's Elementary Number Theory, Section 3 on Linear Diophantine Equations, he poses the problem "A man bought a dozen pieces of fruit--apples and oranges--for 99 cents. If an apple costs ...
k endres's user avatar
1 vote
1 answer
50 views

When formulating the general solutions to a linear diophantine equation $ax + by = c$, does it matter which term is $x$ and which is $y$?

I saw the problem I am working on posted some years ago, and I cannot see how changing the order of the terms can matter, but when applying the general solution after finding one solution, my ...
k endres's user avatar
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1 answer
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Computational Difficulty with Solving System of Equations in an Optimization Problem with 3 constraints (2 active & 1 inactive)

I was given the following problem: Minimize: $$f = -2x + 3y^2$$ Subject to: $$g_1 = (x-1)^2 + y^2 > 1 \\g_2=(x-1)^2 + y^2 \leq 4\\g_3 =x \geq 0 $$ Currently, I am trying to find a minimizer ...
Kakakat's user avatar
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Eliminate the parameters in a symmetric linear-algebra condition

Let positive vectors $x_1,x_2,x_3,p_1,p_2,p_3\in\mathbb R^3_+$. Consider the following condition on the six points $x'$s and $p'$s. Condition: For all $y_1,y_2,y_3$ satisfying the following five ...
High GPA's user avatar
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1 vote
1 answer
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Number of solutions to underdetermined system of equations in modular arithmetic vs real or complex valued equations

I just watched this video about solving a video game puzzle using matrices defined over the integers mod 3, which essentially ended up being a lesson about how the usual rule, that square matrices ...
Mikayla Eckel Cifrese's user avatar
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31 views

can anyone tell me what this symbol imply [closed]

It has greater than symbol with less than symbol Image link
Aarav Raj's user avatar
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1 answer
55 views

What numbers are written on the board? System of equations problem.

Four positive numbers are written on a board. Nika selected 3 of them, calculated their average, and then added the 4th number to the calculated average. She did this in 4 different ways and obtained ...
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Finite operators to express infinite numbers of operators with special conditions

Suppose that we have an infinite number of $\mathbb{C}-$linear operators $\{A_0,A_1,\cdots\}$ where $A_i:V\to W$ and $i\in \mathbb{N}$. If $\dim V$ and $\dim W$ are both finite dimensional and we ...
fusheng's user avatar
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1 vote
1 answer
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Solve the system of equations $x(x+y) - 3 = y$ and $x^2 + y^2 = 5$

The second equation is straightforward: circle, but how do we link this equation with the first? It is a system from the entrance examination in university. The list of topics for this exam is limited ...
Firefly's user avatar
  • 13
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1 answer
34 views

Solving for a Specific Variable in an Underdetermined System of Linear Equations

I have an underdetermined system of linear equations of the form $Ax = b$, where: $$ A = \begin{bmatrix} a_{20} & a_{10} & a_{00} & 0 & 0 \\ 0 & a_{21} & a_{11} & a_{01} &...
Omid Abasi's user avatar
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0 answers
52 views

Solving a mixed system of 2 cubic and 2 quadratic equations with 4 unknowns

I tried plugging these cubic and quadratic equations into Wolfram Alpha and Symbolab but both said the same thing, too much computing time required. Now I am struggling to solve these equations and I ...
Kyle Liu's user avatar
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1 answer
56 views

Why is omitting a column that is a linear combination of other columns not considered an elementary operation in solving a system of linear equations? [closed]

Given a system of linear equations represented by the matrix equation $A\mathbf{x}=\mathbf{b}$, where $A$ is an $m \times n$ matrix. Why does removing a column from $A$ that is a linear combination ...
p0vi3's user avatar
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0 answers
8 views

Connection between Liouville's formula and stability of linear systems

In my university textbook I have a statement: For a linear differential system $\frac{dx}{dt}=A(t)*x$ to be stable is necessary that $\int_{0}^{t} SpA(\tau)d\tau <= M$, where $M$ is a constant, $...
Snork Maiden's user avatar
3 votes
2 answers
121 views

Solving the system $x^4+y^4+z^4=a$, $xy+xz+yz=b$, $xyz=c$

I am trying to solve the following system of equations: $$ \begin{cases} x^4+y^4+z^4=a\\[4pt] xy+xz+yz=b\\[4pt] xyz=c\end{cases} $$ where $a$, $b$ and $c$ are constants and $x$, $y$ and $z$ are the ...
user1331033's user avatar
1 vote
1 answer
24 views

How are the expressions for $\mu$ and $\beta$ obtained?

Given $E^{(p)} = \frac{N}{\beta} + \frac{N\mu}{\sqrt{\beta^2 (\mu^2 - 4\mu)}}$ and $n^{(p)} = \frac{N}{\sqrt{\beta^2 (\mu^2 - 4\mu)}}$, how are the expressions for $\beta$ and $\mu$ obtained in terms ...
KZ-Spectra's user avatar
0 votes
3 answers
79 views

Can we find the root of this equation

Give an equation below: $$ \frac{x^k-a^k}{x-a}=c \qquad (1) $$ where $1<a<x$, $0<k<1$, and $c>0$. I can easily find the numerical root of (1) by using Newton's method or the other tools....
Tyke's user avatar
  • 159
3 votes
1 answer
185 views

Graph of the functions for a system of two equations

I'm a master student in Economics, and I'm going through a paper which involves some mathematics. I'm gonna try to make the argument as far as possible from economics. Consider the following system of ...
Maximilian's user avatar
1 vote
1 answer
68 views

Solve ill-conditioned linear systems

Consider the following linear system of equations : $$Ax = b$$ When solving this system using MATLAB, I found that the condition number of matrix $A$ is extremely large, indicating that the system is ...
Elliot's user avatar
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0 answers
58 views

Solving linear system of equations with constraints on unknowns

I would like to solve a system of linear equations $y=Uh$ for an unknown vector $h$, where I have a few constraints on some of the elements of $h$. Let us consider a small example: $y$ is a vector of ...
Neuling's user avatar
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0 votes
1 answer
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Solving a System of Equations Involving Complex Variables and Their Magnitudes

I’m trying solving a system of equations with two complex variables, x and y . The equations are given by: $$ x + y = c_1 \\ |x| + |y| = r $$ where $c_1$ is a given complex number and r is a ...
Kobamschitzo's user avatar
0 votes
1 answer
56 views

Solving a system of coupled differential equations using eigenvalues and eigenvectors

I have the following the system of coupled differential equation $$ \frac{d}{dt}\begin{bmatrix} y \\ z \end{bmatrix} = \begin{bmatrix} -2g & -\Delta \\ \Delta & 0 \end{bmatrix}\begin{bmatrix}y ...
CauchySchwarzMan's user avatar
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0 answers
12 views

Closed formula for a self-expanding quantity (whose rate of growth per year gradually becomes either 1.0001 or 1.999)

Inspiration for this question (So feel free to skip, because this is not relevant to the math): Chapter 25 of Capital (https://www.marxists.org/archive/marx/works/1867-c1/ch25.htm#S2), where Marx says ...
Zachary Katz's user avatar
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0 answers
19 views

A function of specific form passing through two given points

Let $$s(t; a_0)=a_{0}t^{2}\left(\frac{1}{2}-\frac{t}{3T(a_0)}\right)$$ with $T(a_0)=\sqrt{\frac{6d}{a_{0}}}$ (where $d$ is some positive real constant). Then, let $$ s^*(t; a_0, t_w) = s\left(\frac{t-...
Airat Valiullin's user avatar
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0 answers
34 views

Insights for Outcome Function Involving Multiple Interdependent Variables

I am working on a model involving multiple interdependent variables and systems of equations, and I am trying to gain insights into the behavior and properties of a specific outcome function. Despite ...
blizzard16's user avatar
0 votes
1 answer
36 views

How do I solve this system of PDE'S?

I am having trouble solving this system of PDE's that I got from a big problem $$\partial_\phi B(\phi) + \sin^2(\theta) \partial_\theta C(\theta, \phi) = 0$$ $$\partial_\phi C(\theta, \phi) + B(\phi) ...
some_math_guy's user avatar
0 votes
1 answer
39 views

approximate solution of polynomial equation

I am trying to solve the Following equation for r, $$2 a Q^4+5 r^4 \left(3 c (\omega +1) r^{1-3 \omega }-2 r (r-3 M)-4 Q^2\right)=0$$ Clearly this is unsolvable. But if we substitute a=0 and c=0, the ...
Debojyoti Mondal's user avatar
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0 answers
25 views

Given a matrix $A$ over $F_2[x_1,...,x_n]$ where each entry of $A$ is linear, find the point $x \in F_2^n $ so that $A$ evaluated at $x$ has min rank

Given a matrix $A$ over $F_2[x_1,....,x_n]$ where each entry of $A$ is a linear combination of $x_1,...,x_n$. We say that $A$ evaluated at $x \in F_2^n$ is the matrix over $F_2$ obtained by setting $...
Sander's user avatar
  • 383
0 votes
0 answers
38 views

Finding a basis for an unknown-weights-and-balance puzzle?

I have a collection of unknown integer weights $w_1, \ldots, w_n$. I have a balance which I can use to weigh some pile of weights against some other pile to see which is heavier. Suppose I've done $n$...
user326210's user avatar
  • 17.7k
6 votes
6 answers
622 views

How to Solve a Linear System of Equations with Absolute Values

I have encountered a system of linear equations that involves absolute values: \begin{align} |x + y| &= 1 \\ |x| + |y| &= 1 \end{align} I am having trouble finding resources or methods to ...
Kuly14's user avatar
  • 161
3 votes
0 answers
66 views

Prove $Max(L_1,L_2,L_3)\neq L_2$:$L_i=\frac{S_i^2}{n_i}+\frac{(S-S_i)^2}{n-n_i}$;$S=\sum S_i$;$n=\sum n_i$;$\frac{S_i}{n_i}>\frac{S_{i-1}}{n_{i-1}}$

I will state my question first, and after that, I will write how I arrived to it. You do not really need to see how I arrived to the question, but I just thought it would be rude not to explain that. ...
Francesco's user avatar
0 votes
0 answers
24 views

Existence of a linear equation in n variables

I have a question regarding a linear system that contains a single linear equation in n variables, that is, in the following form: $$a_1x_1 + a_2x_2 + \cdots + a_nx_n = b$$ Where $a_ 1, \cdots, a_n$ ...
studiare's user avatar
1 vote
1 answer
47 views

How to solve, or quantify solutions of, polynomial equations in $\mathbb{F}_2[x,y]/\langle x^\mu - 1, y^\nu - 1\rangle$? [closed]

Suppose I was given an equation in $\mathbb{F}_2[x,y]$ under the identification $x^\mu = 1$ and $y^\nu = 1$ for some integers $\mu,\nu$, with some unknowns $c[x,y]$ and $d[x,y]$. For example: \begin{...
JoJo P's user avatar
  • 133
3 votes
2 answers
152 views

Maximise $xyz$ such that $x+xy+xyz=1$, $y+yz+xyz=2$, $z+zx+xyz=4$

$x, y, z$ are real numbers which satisfy the following: $$x + xy + xyz = 1$$ $$y + yz + xyz = 2$$ $$z + zx + xyz = 4$$ Then find the maximum value of $xyz$. I tried adding and subtracting a few ...
Sujal Motagi's user avatar
1 vote
1 answer
76 views

Solving the system $a^x=(x+y+z)^y$, $a^y=(x+y+z)^z$, $a^z=(x+y+z)^x$, for $x$, $y$, $z$

Given system of equations: $$ \left\{ \begin{array}{c} a ^ x = (x + y + z)^y \\ a ^ y = (x + y + z) ^ z \\ a ^ z = (x + y + z) ^ x \end{array} \right. $$ find the values of $x$, $y$ and $z$ where $...
Samiksha's user avatar
0 votes
1 answer
28 views

Intersection of two vector parametric equations of lines [closed]

...
computermarty's user avatar
0 votes
2 answers
108 views

Solving the system $\cos(x)-\cos(x+y)=0$ and $\cos(y)-\cos(x+y)=0$

At some point when dealing with a problem I had to solve the following system $$\begin{cases} \cos(x)-\cos(x+y)=0\\ \cos(y)-\cos(x+y)=0 \end{cases}$$ I seem to have problems with doing so, or maybe ...
Math Student's user avatar
  • 5,352
1 vote
1 answer
66 views

Count number of possible combinations of $\sum_{i=1}^{n} a_i \leq 10$

If I have $a_1+a_2 \leq 10$, with $a_1, a_2 \in \{0, \, 1, \, 2, \, \cdots, \, 10 \}$: To count the number of possible combinations for $a_1$ and $a_2$ such that $$a_1+a_2 \leq 10\quad\mbox{and}\quad ...
Liszt Morero's user avatar
2 votes
0 answers
39 views

Systems of equations involving unison and intersection

While studying systems of linear equations in linear algebra, i see that the general definition is that: over a field F a standard system ...
thatpithere's user avatar
1 vote
0 answers
21 views

invertibility of overcomplete system of non linear equations

I am currently working on a research problem involving 8 nonlinear equations in 5 variables. While these variables are all real, the equations themselves are complex in general. A colleague has ...
Kobamschitzo's user avatar
1 vote
2 answers
65 views

how to prove $\forall a_n \in \mathbb{R}, n\in \mathbb{N} \exists x \in \mathbb{R} : \sum_{k=1}^n a_k \left( x^k-\frac{1}{k+1} \right)=0 $?

I tried to prove that $\forall a_n \in \mathbb{R} , n\in \mathbb{N}$ then there exist a real root $x$ such that $$ \sum_{k=1}^n a_k \left( x^k-\frac{1}{k+1} \right)=0 $$ for example if $n=2$ and $a_1=...
Faoler's user avatar
  • 1,477
1 vote
2 answers
38 views

Does $A^t$ has no solutions for $A^tx=b$ if given that $Ax=b$ has no solution

If $Ax=b$ has no solution for a $n\times n$ matrix then then can we say $A^tx=b$ also does not have any solution? Here $rank(A)<n$ since the system of linear equation is inconsistent and therefore ...
Akina's user avatar
  • 39
-2 votes
2 answers
40 views

simultaneous non linear equations [closed]

$x^3y^3(x^3+y^3) = 905$ and $x^4y^4(x+y) = 810$. Find values for x and y. Divide the first equation by the second equation, and factor to give $(x+y)^2 = \frac{25}{6}$. If I had stopped at this point, ...
Bob's user avatar
  • 31
2 votes
1 answer
41 views

Is the "box convolution" a linear constraint?

I have a system of linear inequalities $\mathbf{A}\cdot \mathbf{x} \leq \mathbf{b}$; we'll denote the set of solutions by $F$. I want to consider the superset $F^*\supseteq F$ defined by $\{\mathbf{y} ...
user326210's user avatar
  • 17.7k
0 votes
0 answers
65 views

How to solve this system of algebraic equations?

I was solving a Physics Problem and after going on solving I arrived at these equations $$(D_1u+1)\left[{D_2}\cdot\dfrac{(D_1-L)u+1}u+1\right] =\dfrac 1{M_1} $$ $$(D_2u+1)\left[{D_1}\cdot\dfrac{(D_2-L)...
user1318878's user avatar
4 votes
2 answers
63 views

Understanding the schwarz integrability condition for a linear system of pdes.

I have a system of (homogeneous, linear, second-order) equations that looks like: \begin{align*} u_{xx} &= A(x,y) u_x\\\\ u_{xy} &= B(x,y) u_x + C(x,y) u_y\\\\ u_{yy} &= D(x,y) u_y \end{...
user326210's user avatar
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