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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3answers
44 views

Solve the System of Equations in $x$ and $y$

\begin{equation} x+\frac{3\,x-y}{x^2+y^2}=3 \tag{1} \end{equation} \begin{equation} y=\frac{x+3\,y}{x^2+y^2} \tag{2} \end{equation}
1
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1answer
25 views

What is the canonical basis of a dualspace in $\mathbb{R}^3$?

I have the following: Consider the basis $$B := \{\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} \}$$ of the ...
0
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3answers
29 views

Choose h and k such that the system has a solution, a unique solution and many solutions.

Im learning linear algebra, and im tasked with choosing $h$ and $k$ such that this system: $$ \begin{cases} x_1+hx_2=2\\ 4x_1+8x_2=k\\ \end{cases} $$ Has (a) no solution, (b) a unique solution, and ...
0
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1answer
23 views

Solve Coupled System of Equations via Matrix

I have a coupled system of three equations that I am trying to solve via matrices and I am having trouble figuring out how to write out my matrices. My three equations are as follows: $-sx+sy=0$ ...
0
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1answer
19 views

Can someone please provide an intuition behind cramer's rule?

See question. I usually get concepts like this very quickly (no studying required), but this one looks like Chinese. Can someone please help me understand a brief intuition behind Cramer's rule for ...
0
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4answers
33 views

Solving a logarithmic system of equations

I am working on a test study guide and I can't seem to get the correct answer for this system of equations: \begin{align*} \ln(x) &= 3\ln(y) \\ \ 3^x &= 27^y \end{align*} I'm not ...
0
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1answer
12 views

what are some typical systems of equations generating from practical problems?

I want to know some typical forms of system of equations generating from practical problems in engineering/economics/physics,etc. Some examples or research articles would be good. Specifically, I am ...
1
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2answers
14 views

$3$ lines $4$ variables linear equation gaussian

So I'm currently taking a Linear Algebra class and am stuck on a problem. I have the equations: $$\begin{cases}\begin{align}&x + 2y - z + 3t = 3\\ &2x + 4y + 4z + 3t = 9\\ &3x + 6y - z ...
5
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2answers
118 views

How find this system $a^2+b^2=3,a^2+c^2+ac=4,b^2+c^2+\sqrt{3}bc=7$

Find the this system real solution $$\begin{cases} a^2+b^2=3\\ a^2+c^2+ac=4\\ b^2+c^2+\sqrt{3}bc=7 \end{cases}$$ I think that one can use Geometry to solve this system. Maybe there exist an ...
2
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2answers
36 views

System of differential equations using substitution

Exact problem statement Solve the system $\left\{\begin{matrix} x_{1}'(t)=3x_{1}(t)-2x_{2}(t)+e^{2t},x_{1}(0)=a & \\ x_{2}'(t)=4x_{1}(t)-3x_{2}(t),x_{2}(0)=b & \end{matrix}\right.$ by using ...
2
votes
2answers
27 views

Find all solutions to this system of congruences

$$x \equiv 11 \pmod{84} $$ $$ x \equiv 23 \pmod{36}$$ I have the bulk of the work done for this; $x=11+84j$ $x=23+36k$ $\Rightarrow 11+84j \equiv 23 \pmod{36}$ $\Rightarrow 84j \equiv 12 \pmod{36}$ ...
1
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2answers
39 views

Find an integer $x$ satisfying the congruence:

$$x \equiv \ 1 \pmod3$$ $$x \equiv \ 2 \pmod5$$ $$x \equiv \ 8 \pmod{11}$$ From the first, I have $x=3k+1$, $x=5j+2$ from the second and $x=11l+8$ from the third. Subbing the third into the second I ...
2
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2answers
45 views

Describe all integers a for which the following system of congruences (with one unknown x) has integer solutions:

$$x\equiv a \pmod {100}$$ $$x\equiv a^2 \pmod {35}$$ $$x\equiv 3a-2 \pmod {49}$$ I'm trying to solve this system of congruences, but I'm only familiar with a method for solving when the mods are ...
0
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1answer
25 views

Methods for solving linear systems

This is such a basic topic but there are so many different methods proposed for solving a linear system of equations. I recently found a very good source but couldn't really make sense of all the ...
0
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3answers
39 views

Creating and solving large systems of equations

I am trying to follow a solution in a book so that I can build my own model. They produce the set of equations below. The book claims it to be a system of equations with 10 unknowns; however from my ...
0
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1answer
12 views

Systems of Linear equations (substitution method) *Got Part A*IDK about Part B*Part C i have no clue?*

Part A: write the equation that represents M................. y=2x-5 Write the equation that represents N.............................. y=3x+2 ( that is right^^^^^) Part B:using the equations you ...
0
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0answers
17 views

How to solve an overdetermined linear system given equations with different uncertainties

Please, I would like some help to solve the following problem: I have an overdetemined system of linear equation and want to minimize overall error. Up to now, not a problem, I could use least ...
0
votes
1answer
13 views

If M = S, how to isolate a?

So I have to isolate $a$ in $M=S$ $M=1+\dfrac{a}{b}$ $S=a+b$ So, I put it up like this: $1+\dfrac{a}{b}=a+b$ ... right? But then what?
0
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1answer
31 views

Periodic system

We have the following system: $\dot{x}=x-y-x(x^2+y^2)$ $\dot{y}=y+x-y(x^2+y^2)$ Determine the equilibrium points Show that this system has a periodic solution. Use the following substitution ...
0
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0answers
8 views

I have this question: might the sign of the first lyapunov coefficient be different when once solve with the formula and another in matcont. [on hold]

i explain this: suppose we have this ODE system $\dot x_i=f(x_i)$ when i solve it with this formula in Kuznetsov's book with $l_1(0)=(1/2w_0) Re[<p,C(q,q,\bar q)>...]$ the sign be negative, ...
1
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2answers
56 views

A system of nonlinear differential equations

We have the following system in $\mathbb{R}^{2}$ $$\dot{y}_1=2-y_1y_2-y_2^2$$ $$\dot{y}_2=2-y_1^2-y_1y_2$$ i) Calculate the equilibrium points en determine their stability. ii) Draw the Phase ...
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2answers
54 views

find all solutions… i need help for this problem

Find all solutions for the system of equations : where $x,y,z$ are positive integers $$ x^3-y^3-z^3=3xyz $$ $$ 2(y+z)=x^2 $$
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3answers
35 views

Orthogonal vectors and linear systems

Let us suppose we want to solve, with respect to x, the following equation $\mathbf{a}^\intercal\mathbf{b}\;x = 0$ where $\mathbf{a}, \mathbf{b} \in \mathbb{R}^{n} \setminus \{ 0 \}$. It seems clear ...
2
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2answers
31 views

Determining the necessary values for a matrix' coefficients to achieve a certain rank.

I'm having a headache with this... Given the augmented linear system matrix: $$A = \begin {cases} 1 & 0 & 0 & 2 \\ 0 & a-2 & 0 & 0 \\ 0 & 0 & b + 1 & c \\ 0 ...
1
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1answer
55 views

Complex numbers system of equations problem with 5 variables

Let $z_0$,$z_1$,$z_2$,$z_3$ and $z_4$ such that $z_i\in C$ that hold: $$(1)|z_0|=|z_1|=|z_2|=|z_3|=|z_4|=1$$ $$(2)z_0+z_1+z_2+z_3+z_4=0$$ $$(3) z_0z_1+ z_1z_2+z_2z_3+z_3z_4+z_4z_0=0$$ Prove that ...
0
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0answers
62 views

Geometry aspect of a extreme value problem

In a plain with orthogonal coordinate $XOY$, set point $A(a,a)$, and $P$ is a point in function $y=\frac{1}{x}$,where $x>0$. If the distance between $P$ and $A$ is $2\sqrt{2}$.Find all $a$ ...
1
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1answer
36 views

Find the values of $k$ that make this system inconsistent, with unique solution, and with infinite solutions.

I've learned to find the solutions to linear systems using Gaussian Elimination. Moving on, I've found a new kind of exercise I hadn't done before: Find the values for $k$ that make this system: ...
0
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2answers
34 views

What did I do wrong with Gaussan Elimination for $\begin {cases} x + 5y + 11z = -5\\ 2x + 3y + 8z = 4\\ -x + 2y + 3z = -9 \end {cases}$?

Having problems with this one using Gaussian Elimination. Find the solutions for the linear equation system: $$\begin {cases} x + 5y + 11z = -5\\ 2x + 3y + 8z = 4\\ -x + 2y + 3z = -9 \end ...
0
votes
1answer
21 views

Differential Equations - Method of Undetermined Coefficients for products of polynomials and sines

Consider $y''+y= 2x \sin (x)$ I have the solution for the homogeneous equation. Now i am trying to guess a particular solution for: $2x \sin (x)$ My first guess was: $(Ax+B) \cos x + (Cx +D) \sin ...
4
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3answers
43 views

Invalid subtraction when solving system of equations?

I'm trying to solve these two equations: $$\begin{cases} 1-4x(x^2+y^2)=0 \\ 1-4y(x^2+y^2)=0 \end{cases}$$ and I tried to do it by subtracting the first equation from the second, yielding ...
0
votes
0answers
39 views

Solving equation for powers

I would like to find $\gamma$ in: $$ \sum_{i=0}^n x_i^\gamma = y $$ where $n$, $0 \leq x_i \leq 1$ and $0 \leq y \leq n$ are known. Also, $n$ can be fairly large (i.e. from a few thousands to a few ...
0
votes
5answers
40 views

How to solve this homogeneous system, with a missing column?

Find the solution set of triplets $(x,y,z)$ that fulfil this system using Gauss-Jordan: $$\begin {cases} -x + 2z = 0\\ 3x - 6z = 0\\2x - 4z = 0\end {cases}$$ First of all, I don't see any ...
1
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0answers
25 views

Using Gauss-Jordan for an infinite-solutions system

I'm starting to get the hang of this Gauss-Jordan stuff - well, I have never done a system with infinite solutions, so I decided to try this one. You can scroll to the bottom instead to see my doubts ...
1
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4answers
60 views

Are there no solutions for $\begin {cases} 2x+4y = 6\\ 3x+6y = 5\end {cases}$?

I'm trying to solve an equation system using Gauss-Jordan. $$\begin {cases} 2x+4y = 6\\ 3x+6y = 5\end {cases}$$ So, first, the augmented matrix: \begin{bmatrix} 2&4&5\\ 3&6&6\\ ...
2
votes
1answer
36 views

Can all equation systems be reduced to the identity matrix?

I'm trying to learn about solving equation systems using the Gauss-Jordan method. So, you have to convert the equation system to a matrix, and then reduce it to the identity. When you transform it to ...
0
votes
1answer
13 views

Intersection of linear and quadratic functions

I've been stuck on some math work and I'm not sure how to do it. It involves finding the point where a quadratic and linear function intersect only once. Determine the value of $k$ such that $g(x) = ...
1
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2answers
57 views

linear differential equation problem [closed]

Consider the following system of linear differential equations: $$\begin{split} \frac{dx}{dt}&=−3x+y\\ \frac{dy}{dt}&=x−3y \end{split}$$ Find the eigenvalues and eigenvectors associated ...
5
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2answers
32 views

Number of solutions for a system of polynomial equations

Consider the given system of polynomial equations, where all the coefficients are in $\mathbb{C}$: $$\begin{cases} y^n=P(x)\\ Q(x,y)=0\end{cases}$$ I would like to establish that either this system ...
0
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0answers
19 views

perturbation solution of two singular ODEs

I need some help with solving the following system of ODEs: $$\epsilon \frac{dx}{dt}=Ay +ABx(1-y)$$ $$\epsilon\frac{dy}{dt}=Bx(1-y)-y-\epsilon y$$ I'm confused by the fact that both equation are ...
0
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0answers
32 views

Linearization of coupled differential equations

Consider the following coupled differential equation: $\dfrac{\partial x(t)}{\partial t}=(a+y(t))x(t)+b(m_1(t)+m_2(t))y(t)$ $\dfrac{\partial y(t)}{\partial t}=cy(t)+d(m_1(t)+m_2(t))x(t)$ $a,b,c,d$ ...
1
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2answers
42 views

Real world situation with System of Equation with 3 variables?

Where do you run into a real world situation involving 3 variables and 3 equations? Can someone think of a specific example from business, etc? I recall taking an operations research course that ...
3
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1answer
63 views

Solve: $\frac{d^2x}{dt^2}+\Bigl(\frac{d^2y}{dt^2}\Bigr)^2=0$

We have the following coupled Diferential equation: $\dfrac{d^2x}{dt^2}=-\left(\dfrac{y}{x}\right)^2$ $\dfrac{d^2y}{dt^2}=\dfrac{y}{x}$ Then find the solution $x$ and $y$ in terms of $t$ . What we ...
1
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1answer
28 views

Condition for infinitely many solutions

$$\left( \begin{array}{ccc} 2 & 1 & -4 \\ 4 & 3 & -12 \\ 1 & 2 & -8 \end{array} \right) \left( \begin{array}{ccc} x \\ y \\ z \end{array} \right) = \left( \begin{array}{ccc} ...
2
votes
3answers
56 views

Solving Two Equations and Solving

I'm not sure what this is called, but I'll write the problem below. If anyone can help me, as well as tell me where I can get a review of this topic I'd appreciate it. $${\left\{{\begin{align}&2x ...
2
votes
2answers
60 views

Does there exist a polynomial function for every n points, whose extremas are these points?

Given $ n $ points in $ \mathbb{R}^2 $, does there exist a polynomial function of any degree, whose extremas include these $ n $ points? Given 3 points: $ P_1 = (0,4), P_2 = (2,2), P_3 = (4,7) $ And ...
1
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3answers
70 views

$ \frac{x}{4 \ \sqrt{x^2+1}} \ = \ \frac{y}{5 \ \sqrt{y^2+1}} \ = \ \frac{z}{6 \ \sqrt{z^2+1}} $ and $ \ x+y+z \ = \ xyz \ $

Consider the system of equations in real numbers $ \ x,y,z \ $ satisfying $$ \frac{x}{4 \ \sqrt{x^2+1}} \ = \ \frac{y}{5 \ \sqrt{y^2+1}} \ = \ \frac{z}{6 \ \sqrt{z^2+1}} $$ and $ \ x+y+z \ = \ ...
0
votes
2answers
43 views

Conditions of the system of equations.

Find m to the equation:$$\left\{ \begin{array}{l}2x^3-\left(y+2\right)x^2+xy=m\,\,(1)\\x^2+x-y=1-2m\,\,(2) \end{array} \right.$$have experience My try: From $(1)$ and $(2)\,\Rightarrow $: ...
1
vote
0answers
29 views

Lanchester's war model optimization.

Suppose the Lanchester's war model: $f'(t)=-0.5g(t)+x\sin^2(t)$ $g'(t)=-0.5f(t)+\cos^2(t)$ with $f(0)=g(0)=2$. How to estimate how small $x$ can be in order to make $f(t)$ won't reach $0$ on the ...
1
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3answers
43 views

Find three positive numbers $x$, $y$, $z$ whose sum is $10$ such that $x^2y^2z$ is a maximum

I'm self learning from the Vector Calculus book available online in PDF form (page 88). The question is: Find three positive numbers $x$, $y$, $z$ whose sum is $10$ such that $x^2y^2z$ is a maximum. ...
0
votes
1answer
36 views

How can I find four colors with maximum equal difference?

I need to find four colors, expressed as triple $(r_i, g_i, b_i)$ where $0 \le r_i,g_i,b_i \le 1$, $0 \le i \le 3$. Define color difference as $D_{i,j}=\sqrt{(r_i-r_j)^2+(g_i-g_j)^2+(r_i-r_j)^2}$. ...