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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0
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1answer
17 views

How to transform a coupled differential equation into a system with diagonal linear part

Consider the system given by $$iu_t +u_{xx}+2|u|^2u = -v+iu$$ $$iv_t +u_{xx}+2|v|^2v = -u-iu$$ I am trying to transform the system into a system with diagonal linear part. I can solve a problem like ...
1
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0answers
21 views

How to sole this for n? [duplicate]

I started with a hard and unsuccessful way by putting the coefficcients into the matix. I need to solve the system of equation for any natural number n. system
1
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2answers
59 views

How to solve for any given natural number n?

I started with hard way of putting the coefficients into a matrix. But, iz did not help. the following system of linear equations: system
-2
votes
1answer
88 views

Matrix equation with transpose [closed]

How can I solve this matrix equation for $X$: $$ (A^T)X = B(X-Y)C, $$ where $A^T$ is the transpose of $A$. Here, all matrices are small (e.g., $2\times2$). I am especially interested in the following ...
-2
votes
1answer
30 views

Monetary question. [closed]

You have 120 papers of dollars. The 10 dollar bills are 10 times the 5 dollar bills, and the others are 100 dollar bills. How many dollars you have ? I tried to set vriables for the 10 dollar bills ...
0
votes
2answers
63 views

Finding all left inverses of a matrix

I have to find all left inverses of a matrix $A = \begin{bmatrix} 2&-1 \\ 5 & 3\\ -2& 1 \end{bmatrix}$ I created a matrix to the left of $A$, $\begin{bmatrix} a &b &c \\ ...
0
votes
1answer
26 views

Determine the number of solutions of a polynomial system

I have a system of polynomial equations, whose polynomials are all multi linear with coefficients equal to 1. I want to understand if the system has or not real solutions, and if yes, if they are a ...
0
votes
2answers
40 views

Prove that: $\begin{cases} f(\theta_1)\cos \theta_1=f(\theta_2)\cos \theta_2 \\ f(\theta_1)\sin \theta_1=f(\theta_2)\sin \theta_2 \end{cases}$

Let $f(\theta)$ be a continue function for $\theta\in[\theta_1,\theta_2]$. Prove that: \begin{cases} f(\theta_1)\cos \theta_1=f(\theta_2)\cos \theta_2 \\ f(\theta_1)\sin \theta_1=f(\theta_2)\sin ...
0
votes
3answers
25 views

Necessary condition for uniqueness solution in a system of non-linear equations

Consider a system of non-linear equations with $n$ equations and $m$ unknowns. Is $m=n$ a necessary condition for having one unique solution?
0
votes
1answer
19 views

proof of existence of solution to componentwise inequality using only linear algebra

For $A \in \mathrm{Lin}(\mathbb{R}^n, \mathbb{R}^m)$ , $m=n$ and $A$ is invertible, I am able to prove the existence of a solution $x$ such that $A x \succeq 0$ where $\succeq$ denotes componentwise ...
0
votes
0answers
27 views

Which of the following is true about $m \times n $ mayrix of rank $n$.

Let $A $ be an $m \times n$ matrix of rank $n$ with real entries. Choose the correct statement. 1.$Ax=b$ has a solution for any $b$. 2.$Ax=0$ does not have a solution. 3.if $Ax=b$ has a solution ...
1
vote
0answers
38 views

Closed-form solution for a matrix equation involving pseudo-inverses and Frobenius norms

Let $A\in \mathbb{C}^{m \times n}$ be a wide unknown complex matrix ($m<n$), and $\Sigma\in \mathbb{R}^{p \times n}$ a known rectangular diagonal matrix. $A$ has full row-rank. Is it possible to ...
1
vote
2answers
23 views

Split complex system of equations into two real systems

Suppose I have a complex system of equations in 3 unknowns, like this one: $$ \pmatrix{ 40 & -20 & 0\\ -20 & 20-20j & 30+10j\\ 4 & -5 & 1 } \pmatrix{ x_1+j x_2\\ y_1+j y_2\\ ...
0
votes
1answer
104 views

Solutions to simultaneous Diophantine equations $2y^2-3x^2=-1$ and $z^2-2y^2= -1$

I am looking for integer solutions for the following set of equations: $2y^2-3x^2=-1$ $z^2-2y^2= -1$ I know that there are the solutions (1,1,1) and (-1,-1,-1) for this set of ...
0
votes
0answers
33 views

Implementation of Poincaré–Miranda theorem

To test whether a continuous function has one simple root in a given interval $[x0, x1]$ is relatively easy: according to Intermediate value theorem when the sign of function value at $x0$ is opposite ...
0
votes
1answer
37 views

How this type of equation is solved? [closed]

I'm solving a relative and ends when the function (x²+y²)²-4x² derive out this equation, but not that I have to do to get resolve it $$f'x = 4x³+4y²x-8x$$ $$f'y = (4x²+4y⁴)y$$ How solve this ...
3
votes
2answers
55 views

How to solve simultaneous exponential equations with polynomial parts?

I have been puzzled at how to simultaneously solve the following equations and before I give up entirely I thought I'd turn to fellow mathematicians first: $$2^x=3y$$ $$2^y=5x$$ I have graphed both ...
0
votes
0answers
29 views

Algebraic and geometric representation of the linear system

$1)$ Show that for $\forall\alpha\in\mathbb R$ the set of solutions of a system $$x-y+2z-t=1$$ $$2x-3y-z+t=-1$$ $$x+(\alpha-4)z=\alpha-3$$ is not empty. $2)$ Describe that set for all values $\alpha$ ...
0
votes
1answer
27 views

Determine $a,b\in\mathbb R$ such that for linear transformation $f:\mathbb R^3\rightarrow \mathbb R^3$ is valid: $(4,3,4)\in Im(f)$.

Determine $a,b\in\mathbb R$ such that for linear transformation $f:\mathbb R^3\rightarrow \mathbb R^3$ given by matrix $ \begin{bmatrix} a & 1 & 1 \\ 1 & b & 1 ...
0
votes
3answers
35 views

System of Linear Equations with integer Coefficients

Consider the following system of linear equation: \begin{align} 2a + 4b &= a + 3c\\ 2a + 3b &= 4a + 2b\\ 4a + 2b &= b + nc \end{align} for $a,b,c \in \mathbf{R}_{+}$. How do I ...
1
vote
5answers
50 views

Four statements, One statement is false math problem

When trying to recall some facts about the ages of his three aunts, Josh made the following claims: Alice is fifteen years younger than twice Catherine’s age. Beatrice is twelve years older than ...
2
votes
4answers
62 views

Solve for $x$ if $4^{\frac{x}{y} + \frac{y}{x}}$ $=$ $32$ and $\log_3(x+y)+\log_3(x-y)=1$

Question: Solve for $x$ if $4^{\frac{x}{y} + \frac{y}{x}}$ $= 32$ and $\log_3(x+y)+\log_3(x-y)=1$ My attempt: With the first equation $$4^{\frac{x}{y} + \frac{y}{x}} = 32$$ ...
0
votes
0answers
35 views

How to Calculate Uncertainty in Simultaneous Equations

I need to solve the following simultaneous equation in three variables. I am able to do this, but I am unsure as to what the uncertainty in the values are. a + b = 42.6 ± 0.1 a + c = 56.3 ± 0.1 b + ...
0
votes
1answer
26 views

How to write a transfer function (in Laplace domain) from a set of linear differential equations?

Provided I have a system of linear differential equations (in time domain) such as: $$\begin{cases} \dot{x}(t)=Ax(t)+By(t)+Cz(t)\\ \dot{y}(t)=A'x(t)+B'y(t)+C'z(t)\\ \dot{r}(t)=B''y(t)\\ \end{cases}$$ ...
1
vote
1answer
66 views

Solve the system of equations $\begin{cases}x^3-3x=y \\ y^3-3y=z \\ z^3-3z=x \end{cases}$

Find the number of real solutions to the system of equations $$\begin{cases}x^3-3x=y \\ y^3-3y=z \\ z^3-3z=x \end{cases}$$ Let $f(x) = x^3-3x$ then for $x\in \mathbb{R}-(-2,2)$ we have $x_1 ...
0
votes
1answer
36 views

System of modular equations with unknown modulus

I have a linear congruential generator, which works based on this equation: $X_i = (aX_{i-1}+b) \mod m$. I'm trying to compute next number, but everything I have is given output. So for example $\;a ...
1
vote
2answers
39 views

How to solve ${(1 + x)^2}y'' + (1 + x)y' + y = 4\sin \ln (1{\text{ + }}x)$?

I don't know how to solve this equation: ${(1 + x)^2}y'' + (1 + x)y' + y = 4\sin \ln (1{\text{ + }}x)$ When it's homogeneous, I try to solve by power series tediously. Is there any good way to ...
1
vote
2answers
39 views

How to find solutions for $a$ and $b$ where $9 \equiv 4a+b \pmod {26} $ and $10 \equiv 19a+b \pmod {26}$? [closed]

$$9 \equiv 4a+b \pmod {26}$$ $$10 \equiv 19a+b \pmod {26}$$ How can I solve the following system?
3
votes
3answers
61 views

Solving system of three quadratic equations

$$\begin{cases} x^2 = yz + 1 \\ y^2 = xz + 2 \\ z^2 = xy + 4 \end{cases} $$ How to solve above system of equations in real numbers? I have multiplied all the equations by 2 and added them, then got ...
0
votes
1answer
17 views

Simultaneous equations with a parameter

Show that the following system of equations has a solution for any value of the constant $\lambda$, using matrix method. \begin{cases} & x+2y+4z = 4 \\& 2x+3y+6z = 0 \\& ...
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votes
1answer
47 views

solving a system of simultaneous equations with no values

I need to get the conditions on $a_i$ for the set: $$x_1+x_2+x_3=a_1a_2a_3$$ $$x_1+x_3+x_4=a_1a_3a_4$$ $$x_1+x_2+x_4=a_1a_2a_4$$ $$x_2+x_3+x_4=a_2a_3a_4$$ The best I get is ...
0
votes
1answer
23 views

Properly using natural logs to solve for a variable

I'm attempting to take a logistic graph and create an equation using the logistic model of continuous growth. I have taken the equation and simplified it down to ...
0
votes
1answer
17 views

Is $Φ^T$ a linear operator which transforms simultaneous equations such that we obtain LMS solution?

The below explanation is long winded, if you already know about using pseudo inverse to find the best fit solution to a set of simultaneous equations please go down to the tl;dr The Problem Given a ...
2
votes
1answer
25 views

How to calculate solution effciently [closed]

iI have the following equation which I want solve, what is the most efficient way? I could not find closed form solution. $\Sigma$ is a diagonal positive definite matrix. $s, \lambda$ are positive ...
1
vote
1answer
21 views

Classification for linear systems

can someone explain to me the classifications for linear equations? I don't understand what the terms mean. Consistent, inconsistent, and dependent and how a system of equations can be both.
4
votes
1answer
55 views

solve the following system of equations in real $x$, $y$

solve for real $x,y$ $$2^{x^2+y}+2^{x+y^2}=8 \tag{1}$$ $$\sqrt{x}+\sqrt{y}=2 \tag{2}$$ Trivially $x=y=1$ Now Equation $(1)$ can be written as $$2^{x^2+(\sqrt{y})^2}+2^{x+(\sqrt{y})^4}=8$$ so we ...
0
votes
0answers
17 views

How can I convert a system of linear equations to a matrix multiplication and vice versa?

I am learning about homogeneous transformations, and I just saw an example where the author took a system of two linear equations and rewrote it into a matrix multiplication. I am just wondering if ...
0
votes
0answers
36 views

How to solve simultaneous logical equations?

Suppose I have two equations $$y = (x\oplus a)+b \\ x = (y\oplus c)+d$$ All numbers are in base 10. $\oplus$ is the XOR operation for binary numbers. How do I solve such equations manually, or is ...
0
votes
1answer
25 views

Infinite non deviating slope

Okay, I had a question that my math teacher didn't know the answer to, and that I haven't found an answer for on the web. Say you are graphing a system of equations, right, and you have ...
1
vote
0answers
28 views

Can we solve this system of inequalities analytically?

Let $A$ be positive real number and $k$ a positive integer. How to find the analytical solution of this system? Find the $a_i$ \begin{align} \begin{cases} ...
1
vote
1answer
49 views

Is this fact true or how to make it true?

Let $a_1,\ldots,a_n, A$ be positive real numbers and $k$ a positive integer. Is this correct? If we have $$\sum_{i=1}^n\ln\left(1+a_i\right)\geq k\ln\left(1+A/k\right),$$ and $$\sum_{i=1}^na_i\leq ...
0
votes
3answers
30 views

Column vectors as entries of a column vector

I'm having a hard time understanding notation. Because of notation, I've lost hours and hours trying to understand a simple concept. I'm going to post a picture of the pdf that I've so that you can ...
0
votes
1answer
29 views

How to solve nonlinear system generated from $(x_i - x_j)^2 \approx f_{ij}$

Looking for advice/help with solving a nonlinear system generated from the equation: $(x_i - x_j)^2 \approx f_{ij}$ where $\textbf(X) = (x_1,x_2,x_3,...,x_n)$, $f_{ij}$ are known, solve for ...
0
votes
0answers
43 views

Solution of a partial integro-differential equation system

I'm trying to solve the following system: $\frac{d P_0(t,x,y)}{\partial t}+\frac{\partial P_0(t,x,y)}{\partial x}+\frac{\partial P_0(t,x,y)}{\partial y}=-\left(Q_1(t)+Q_2(t)\right) P_0(t,x,y)+\mu _1 ...
2
votes
1answer
32 views

A system of linear PDEs in cylindrical coordinates.

I have the following system of PDEs: $$\partial_r b_2 - \partial_{\theta}b_1 = 0, \\ \partial_{\theta} b_3 - \partial_{z}b_2 = 0, \\ ~~~~~~~\partial_rb_3 - \partial_{z}b_1 = \xi(r, z), $$ where ...
1
vote
0answers
21 views

Can We multiply a constant in a proportionality?

If we have a proportionality stating y is proportional to x can we say y is proportional to maybe 2x or kx in general? Because if y is proportional to x and x in turn is a function say 5z then y is ...
0
votes
1answer
38 views

Systems of Equations (Inconsistent)

Question: Consider the following system of three equations: $$2y+2z=9-2x$$ $$x=12-3y-4z$$ $$Ax+5y+6z=B$$ Find values of A and B which makes the system of equations ...
0
votes
1answer
30 views

Systems Of Equations (Find any other equation in the form $ax + by + cz = k $)

Question: Prove the point $(2,5,-4)$ is a solution to the two equations: $$ x + 2y + 3z = 0 $$ $$ 2x-y-2z=7$$ Find any other equation in the form $$ax + by + cz = k $$ ...
4
votes
1answer
115 views

Does $A$ commute with $e^{\int A \: dt}$

I have been studying the linear system of the form: $$D_tX = AX + \textbf{b}$$ Where $A$ is not necessarily constant Suppose we aim to find an integrating factor $M$ such that: $$M[D_tX - AX] = ...