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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22 views

Simultaneous recurrence relations

Currently working on solving this set of three simultaneous recurrences, but having some trouble. Tried various substitutions, but still cannot seem to make any progress. Also, none of the three ...
3
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1answer
47 views

A System of Infinite Linear Equations

Suppose that $\{a_{i}\}_{i=-\infty}^{\infty}$ with $\sum_{i=-\infty}^\infty a_{i} \lt \infty$ is known and that $\{b_i\}_{i=-\infty}^{\infty}$ is such that $$\sum_{i=-\infty}^\infty a_{i}b_{-i} =1,$$ ...
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4answers
43 views

Symmetric system of $3$ equations

I need help solving the following system for all real ordered triples $(x,y,z)$ without guessing and checking: $$x+y+z=23$$ $$xy+yz+xz=144$$ $$xyz=252$$ Preferably, the solution should use methods ...
3
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1answer
41 views

Solving a system of non linear equations

I have got a system of non-linear equations of the form $$A x_1^B \exp \bigg(\frac{- C}{x_1} \bigg) = k_1$$ $$A x_2^B \exp \bigg(\frac{- C}{x_2} \bigg) = k_2$$ $$A x_3^B \exp \bigg(\frac{- C}{x_3} \...
1
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1answer
34 views

System of two Nonlinear equations

I have a probably very simple problem here. A system of nonlinear equations. $$\left\{ \begin{align} & {{x}^{2}}+{{y}^{2}}=26 \\ & x+{{y}^{2}}=6 \\ \end{align} \right.$$ I started with ...
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0answers
41 views

Incorrect answer - Simultaneous Differential Equations

The questions states solve for y such that $$y' = \begin{bmatrix} -4 & 2 & 1 \\ 1 & -3 & 1 \\ 3 & -3 & -2 \\ \end{bmatrix}y , y(0)= c = \begin{bmatrix} 1\\5\\3 \end{...
1
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2answers
21 views

Dimension of the span of two parallel lines in $R^4$.

I am asked if the following question is true or false: Let $r,s$ be two parallel lines in $R^4$ then the dimension of $Span(r \cup s)$ is strictly less than $3$. I think this is true because two ...
2
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4answers
110 views

Solution of $(n+1)^{1/3}-n^{1/3}=\frac{1}{12}$

Solve the given equation for $n$ $(n+1)^{1/3}-n^{1/3}=\frac{1}{12}$ How to approach this particular question? Sorry cannot show any work because the only approach I can see is take cube on both ...
3
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1answer
73 views

Help required in finding solution to overdetermined system of equations?

I have access to M probability measures, $P_e(c_1),P_e(c_2),\cdots,P_e(c_M)$, defined as \begin{equation} P_e(x) = p(x|y) = p(y|x)\cdot \mathbb{P}(X=x) \frac{1}{\sqrt{2\pi\sigma^2}} \exp\Big[-\frac{(y-...
1
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2answers
30 views

matrix calculation

Let $p= \begin{pmatrix} x & y \\ z & v \end{pmatrix}\in M_2(\mathbb{C})$ such that $p^2=\overline{p}^t=p$ and rank(p)=1. Why is $p=\begin{pmatrix} t & l\sqrt{t(1-t)} \\ \overline{l}\sqrt{...
2
votes
4answers
98 views

Is this possible to solve through algebra?

$$150 \equiv 17 \mod x, \qquad 100 \equiv 5 \mod x $$ Solve the simultaneous equation? Is this even a simultaneous equation? How do I find the value of $x$ too? I was doing a question and came up ...
3
votes
3answers
50 views

If $a-c = 9$ then find the value of $b-d$.

If $a,b,c,d$ are positive integers such that $\log_a b=\frac{3}{2}$ and $\log_c d=\frac{5}{4}$, if $a-c = 9$ then find the value of $b-d$. We get $b=a^{3/2}$ and $d=c^{5/4}$ Hence $b-d=a^{3/2}-c^{...
1
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1answer
27 views

Linearly independent subset - a simple solution?

Problem: Let $\{ v_1, \ldots, v_n \}$ be a linearly independent subset of $V$, a vector space. let $$ v= t_1 v_1 + \cdots + t_n v_n $$ where $t_1, \ldots t_n \in \mathbb{R}$. For which $v$ is ...
0
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1answer
24 views

Solve Graphically

Solve the given systems of equations by graphical method: $$x^2+y^2=5$$ and $$y=2x$$ My Attempt Let's have a look at the second equation ; $$y=2x$$ This is a linear equation in two variables ...
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1answer
29 views

Determining if system is consistent, and if it is determine if the solution is unique

In the following matrices [] is a nonzero entry and ∗ is a entry that may or may not be zero. For each of these (augmented) matrices determine if the associated system is consistent, and if it is ...
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1answer
33 views

Solution of given equations for $x$ and $y$

Solve for $x$ and $y$ $$(2x)^{\log 2}=(3y)^{\log 3}$$ $$3^{\log x}=2^{\log y}$$ Could someone give some hint to approach this question?
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2answers
53 views

Find three different systems of linear equation whose solutions are.. [closed]

Find three different systems of linear equation whose solutions are $x_1 = 3, x_2 = 0, x_3 = -1$ I'm confused, how exactly can I do this?
2
votes
2answers
69 views

How do I solve this System of Equations?

How do I begin to solve this system? $$x^2=y+a$$$$y^2=z+a$$$$z^2=x+a$$ Do I take the square roots of $x,y$ and $z$? If so, we get $$x=\pm\sqrt{a+y}$$$$z=\pm\sqrt{a\pm\sqrt{a+y}}$$$$y=\pm\sqrt{a\pm\...
2
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1answer
83 views

$n$ electric charges on a circle

The following problem is of physical nature, but its core consists of pure mathematics, so I ask it here: Suppose we have $n$ electric charges $q$ on a circle. They can move freely around it, but ...
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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
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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 ...
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2answers
65 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.
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1answer
31 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, ...
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2answers
156 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, ...
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2answers
37 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
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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 ...
0
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 ...
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2answers
25 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 ...
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0answers
25 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
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1answer
34 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
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1answer
33 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
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0answers
35 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?
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1answer
28 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 ...
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3answers
34 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
31 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
64 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 ...
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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
28 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
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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
16 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.
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1answer
65 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
47 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
26 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 = ...
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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 |...
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2answers
33 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
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2answers
28 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
53 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
39 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.