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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35 views

Solving equations, Math olympiad, using vieta relation?

So the question asks to solve for real valued $a$ such that $b,c,d\in\mathbb{R}$ $$abcd=-1$$ $$(a+b)(c+d)=-1$$ $$ac+bd+a+b+c+d=-1$$ $$ab+cd=ac+a+c$$ So assuming the four numbers are roots of a quartic ...
1
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1answer
49 views

This three-variable system of equations seems impossible to solve

$$g = af^b + c$$ $$i = ah^b + c$$ $$k = aj^b + c$$ I want to solve for $a$, $b$, and $c$. $f$, $g$, $h$, $i$, $j$, and $k$ are inputs to the equations, so they don't have to be solved for. Just ...
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0answers
5 views

Using MATLAB to solve a system of 2nd order non linear ODEs

I have 2 coupled non linear 2nd order ODEs which describe a particle's trajectory in space, subject to an initial horizontal and vertical velocity, and also to gravitational and aerodynamic forces. ...
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1answer
22 views

Basic solutions of linear equations

I am struggling to find the bases of these. I have put it in the form Ax=b, however all of the examples in my notes use the formula $A_Bx_B+A_Fx_F=b$ however, this only seems to work for square ...
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0answers
19 views

How to form an equation from given graph

Hello, How can I form the mathematical equation from the given characteristics? I have attached here the excel file data of X andY axis along with the formed graph. I want to form the mathematical ...
2
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3answers
29 views

Finding maximum points by constrain optimization (multivariable calculus)

Find the maximum value of the function $f(x,y)=x^2+y^2+2x+y$, on the closed disc (the circle together with the region inside the circle) of radius 2, centred at the origin. What i tried I know that ...
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1answer
31 views

Analytic solution for a type of PDE systems

Peace be upon you, I have the following system of partial differential equations \begin{align*} \begin{cases} \frac{\partial}{\partial a}S(a,b,c,d)=f_1(a)\\ \frac{\partial}{\partial ...
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1answer
24 views

Tricky logarithm problem

I having a problem in this logarithm problem involving modulus- Solve for x |x-1|^((log(x))^2-2log(x))=|x-1|^3 Bases same so powers equal. If I take log x as a then I get the following quadratic- ...
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0answers
18 views

Prove $x\to 0$ as $t\to \infty$ if we consider the system of equations $x'=(A+B(t))x$ where $B(t)\to 0$ and $A$ has negative eigenvalues.

Consider a matrix $A$ such that all of its eigenvalues are negative. Consider $B(t)$ where $B(t)\to 0$ as $t\to\infty$. Then consider the system of equations $$ x'=(A+B(t))x$$ Prove that any ...
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2answers
44 views

Solve for reals $x, y\in \mathbb R$ given system of two non-linear equations.

Solve for reals:- $$\begin{align} 5x\left(1+\frac{1}{x^2+y^2}\right)& =12\\ 5y\left(1-\frac{1}{x^2+y^2}\right)&=4\end{align}$$ I got this relation $$6x^{-1}+2y^{-1}=5$$ Now I substituted ...
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1answer
30 views

Linear programming and the simplex method

I am trying to solve this system of equations. My approach would be to introduce slack variables and then somehow use the simplex algorithm to solve this. Can anyone show me how this is done?
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1answer
37 views

Solving $Ax=B$: what's wrong with this linear algebra argument?

With $K>L$ and assuming that we are working with real variables, suppose that $B$ is $K\times 1$ and $A$ is $K\times L$ with full column rank. I'm trying to find $x$ ($L\times 1$) satisfying: $$ ...
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1answer
8 views

Scaling variables in homogeneous equation of degree two in a,b,c

The problem I'm having trouble with is: Let $a,b,c$ be nonzero real numbers and let $a^2 - b^2 = bc$ and $b^2 - c^2 = ca$. Prove that $a^2 - c^2 = ab$. The solution strategy given in the course was ...
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1answer
19 views

Solving Yoshida equations

I want to solve $a$, $b$ and $c$ out of the following set of equations \begin{cases} a + b + c = 1 \\ a^{p+1} + b^{p+1} + c^{p+1} = 0 \\ a = c \\ \end{cases} where $p$ is even. But I absolutely have ...
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0answers
12 views

What mechanism can be used to solve these tasks given to a 5th grader?

Basically this table has to be filled with numbers so the expressions in each column and row would be equal to the last element of that column or row, which is given. Is there some mechanism that ...
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0answers
17 views

System solving with Substitution and Matrices

My class was able to produce solutions using Substitution on the following System: $$ \left\{ \begin{array}{c} x+y+z=0 \\ 2x+3y+2z=-1\\ x-y+z=2 \end{array} \right. $$ The solution was: x = 1, y = ...
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0answers
22 views

Best approach to matrix representation of system of nonlinear ODEs

I have this system of ODEs: $$ \frac{dS}{dt}=\pi S-\beta S Z\\ \frac{dZ}{dt}=\alpha S Z - \delta Z $$ I am trying to rewrite them in the form : $$ \pmatrix{\dot{S}\\\dot{Z}}=\mbox{diag}(S,Z) ...
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0answers
34 views

Recovering a kernel from a system of equations

Suppose $f\in C([0,\frac{3}{4}]^2)$ and $$\begin{array}{rlr}\text{i.}& \int_0^{\frac{3}{4}-x} f(x,y)dy=-\frac{1}{2}x^2+\frac{9}{32}&\forall x\in [0,\frac{3}{4}]\\ \text{ii.}& ...
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0answers
42 views

How to solve a system of two differential equations describing the concentration in a leaky tank?

While filling up a chemicals container at a constant rate of 300 litres/min, the crew of a naval ship discover two leakages at the bottom of the container. They discover that the chemical is leaking ...
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1answer
35 views

Find the standard matrix of the linear transformation

Suppose there is a linear transformation $T:\mathbb{R^2} \rightarrow \mathbb{R^2}$ such that $$T\left( \begin{array}{ccc}2 \\ 1 \end{array} \right)=\left( \begin{array}{ccc}1\\ 4 \end{array} ...
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2answers
46 views

Solving the system $a^2-6=2\sqrt{2c+6}, \, b^2-6=2\sqrt{2a+6}, \, c^2-6=2\sqrt{2b+6}$

Question: Solve the following system for $a,b,c\in \mathbb{R}$: $$\begin{cases} b^2-6=2\sqrt{2a+6}\\ c^2-6=2\sqrt{2b+6}\\ a^2-6=2\sqrt{2c+6} \end{cases}$$ I found the following:$$ ...
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0answers
63 views
+100

Solution of parabolic PDE system

For the following parabolic PDE system, $u(x,t)$ and $v(x,t)$ are functions of independent variables $x$ and $t$, $x\in[a,b]$. \begin{equation} \begin{cases} \frac{\partial}{\partial ...
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2answers
54 views

Solving a system of two cubic equations

I'm trying to solve a system of two cubic equations with two variables x and y. The original problem was to solve the equation $z^3=-4i \overline{z}$. I know how to solve it using polar form. Now I ...
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0answers
32 views

Sharkovskii's theorem in two dimensions?

Question A weak form of Sharkovskii's theorem in $1$D dynamical systems states that, if a continuous function $f:I\to I$ does not include a periodic point of least period $2$ on $I$, then ...
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2answers
39 views

Roots of simultaneous power sum equations (numerically or otherwise)

I'm a physicist, and I've come across a problem in my research where I need to solve a set of equations looking like (e.g. in 3D) $$r_1 + r_2 + r_3 = k_1$$ $$r_1^2 + r_2^2 + r_3^2 = k_2$$ $$r_1^3 + ...
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0answers
7 views

Two Column Proof for a System of Linear Equations

Given $8x+10=y$ $\frac{1}{4}y=x$ Prove: The solution to the system of equations is $(\frac{1}{2},6)$ So I have my two columns, and in step 7 I got $x=\frac{1}{2}$, then in 11 I got $y=6$ ...
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1answer
26 views

Mixture problem might be missing something

A winemaker wants to mix a $10\%$ alcohol wine with $20 \text{ kg}$ of a $55\%$ wine to make a $35\%$ wine cooler. How much of the $10\%$ should be used? I started with $.1x+.55y=.35$ then I'm not ...
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2answers
45 views

Evaluate a linear system of three equations

Solve for $x, y\ \text{and}\ z\ $: $x-3z=10\\ -x+y+2z=7\\ 2x+2y-5z=-8$ My working: $$\left(\begin{array}{ccc|c} 1 & 0 & -3 & 10 \\ -1 & 1 & 2 & 7 \\ 2 & 2 & -5 & ...
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1answer
55 views

Rational solution to AX=0

Let $\mathcal{M}_{n,p}(\mathbb{K})$ be the set of matrices $n\times p$ with coefficients in $\mathbb{K}$. Let $A\in\mathcal{M}_{n,p}(\mathbb{Q})$. We suppose there exists a non zero solution ...
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1answer
15 views

Creating a system of linear equations.

An average mark is computed for 100 students in Business, an average is computed for 300 students in Arts, and an average is computed for 200 students in Science. The average of these three averages ...
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2answers
27 views

Solve a system of equations when one is linear and the other is quadratic

$x+y=3m$ $xy=2m^2$, $m$ is the parameter. I came to this $2m^2-3mx+xy=0$. The solutions have to be:$(m,2m),(2m,m)$. But I can't understand what is the role of this parameters, I don't know how to ...
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3answers
34 views

Creating a system of linear equations

A bookkeeper is trying to determine the prices that a manufacturer was charging by examining old sales slips, which show the number of various items shipped and the total price. He finds that $20$ ...
2
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1answer
43 views

Solving systems of linear equations.

I'm just curious at to creating the system of linear equations for the certain conditions given by this question. ...
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1answer
22 views

Solving coupled non-linear equations

I am struggling to understand what the following question requires me to do: I believe I need to differentiate implicitly, but am unsure how I show it cannot be done.
2
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1answer
38 views

Lagrange multipliers subject to a 3-variable, 4th degree constraint function?

I have recently been tackling the following problem: If $a+b+c = 0 $ and $ a^2 + b^2 +c^2 = 1$, work out $a^4 +b^4 +c^4$. Could this problem admit a solution through the method of lagrange ...
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0answers
38 views

Non-linear least squares solver to solve a system of non-linear equations?

Can I use a non-linear least squares solver to find the solutions of a system of non-linear equations? From Wikipedia: "The method of least squares is a standard approach to the approximate ...
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3answers
33 views

System of two equations with two unknowns - can't get rid of $xy$

The system is: $x^2 + 2y^2 + 3xy = 12$ $y^2 - 3y = 4$ I try to turn $x^2 + 2y^2 + 3xy$ into $(x + y)^2 + y^2 + xy$ , but it's a dead end from here. Can anyone please help?
2
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0answers
51 views

Computing a “cheap” upper bound on the norm of the solution to a linear system

Consider the linear system $A x = b$, where $A$ is an invertible, $n \times n$, real matrix. I would like to compute a "cheap" upper bound on the (p-)norm of the solution; i.e. $\|x\|_p$. One can, of ...
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0answers
16 views

final value theorem in the presence of white noise

I apply the final value theorem to get the steady-state error with the presence of white noise and I just keep getting zero. To me, it seems wrong to have zero steady-state error when there is noise ...
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5answers
53 views

Creative way to solve a linear system

Does anyone know a method for solving the following linear system: Here, $\pi_0, \, \pi_1, \, \pi_2$ are the variables. I tried solving by setting up a simple matrix and row-reducing, but it ...
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0answers
41 views

Hopf bifurcation how to prove

I have this system of differential equations: \begin{equation} \frac{dx}{dt}=1-(b+1)x+x^2 y\\ \frac{dy}{dt}=bx-x^2 y \end{equation} I now that we will have a bifurcation when $b$ grows and ...
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1answer
28 views

How to find all positive integer solutions of this tricky system of equations?

What are all tuples off postive integers w,x,y,z that fullfill following system of equations: $$ x + 10z^2 = 2014 $$ $$ 2y + z = 54$$ $$ (y+2x + \frac{7}{2}w)z=1211$$ I am really lost - all my ...
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0answers
17 views

How can we solve a trigonometric system of equations with Fourier Transform?

Today the professor in linear algebra, when asked if there is any other way rather than linearize a trigonometric system of equations by letting variables as the trigonometric functions of each ...
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1answer
19 views

How to solve this system of 3 equations with substitution?

I have the system: $$-4 + λ = -3a + at\\ 1+2λ = -a + at\\ 3λ = 3a-at$$ but whenever I try to substitute, I end up getting lots of fractions that are hard to work with. By summing the 2nd and 3rd ...
2
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0answers
41 views

Math software for plotting phase portraits

I'm looking for math software which is possible to plot phase portraits for ODE and systems of differential equations. Is there a software which can create not only simple 2D phase portrait plots but ...
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3answers
39 views

Simple equations word problem

In a three-digit number, the difference between its hundreds digit and its tens digit is equal to the difference between its tens digit and its units digit. Also the sum of the digits is $9$. How many ...
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2answers
29 views

Huge linear system of equations with powers of $2$

I've got a large system of equations: $$ \begin{align*} (2^0)^na_n + (2^0)^{n-1}a_{n-1} + \cdots + (2^0)^1a_1 &= 4^0 \\ (2^1)^na_n + (2^1)^{n-1}a_{n-1} + \cdots + (2^1)^1a_1 &= 4^1 \\ \vdots\\ ...
0
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1answer
65 views

Simplification of nasty expression

I have the following equation which I am trying to solve, $$ \frac{x^2}{y^2}- \frac{x^2}{y^4} -\frac{1}{2} \leq 0$$ Can anyone think of a way of simplifying the above, I don't think this is a form ...
2
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0answers
34 views

Nontrivial solutions for a system of equations

Consider $t:[0,1]^2\to R$ that is differentiable a.e. and satisfies conditions (i)-(ii): (i) $$ \int_0^1 \frac{\partial t}{\partial t_1}(x,y)f(y\mid x)\,dy=0, \quad \forall x\in[0,1] \\ \int_0^1 ...
1
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2answers
36 views

Property of a system of two inequalities

I have this system $$\begin{cases} a+b>1 \\ a-b>1 \end{cases}$$ can I sum the second inequality to the first getting $a>1$? Or this property can be used only equations?