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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2
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1answer
43 views

Two variable equation

I'm stuck with the following example (42.). Some help is much appreciated. Thank you.
1
vote
0answers
38 views

Can trigonometric equations be graphed?

I was solving various trigonometric equations. I was confused that how are they solved easily by using methods that are useful to solve algebraic equations. Do the trigonometric functions in ...
0
votes
3answers
39 views

Solve the Equation.

$$ \begin{bmatrix} 1 & 1 \\ 1 & 1 \\ \end{bmatrix} \begin{Bmatrix} v_1 \\ v_2 \\ \end{Bmatrix}= \begin{Bmatrix} 0 \\ 0 ...
0
votes
1answer
52 views

System of equations and the Brouwer's Fixed-Point Theorem.

Let's consider the following system of equations: \begin{eqnarray}{ e^{xyz} = \frac{x}{\sqrt{e^{2xyz}+1}}\\ \cos(x+y+z) = \frac{y}{\sqrt{e^{2xyz}+1}}\\ \sin(x+y+z) = \frac{z}{\sqrt{e^{2xyz}+1}} ...
0
votes
0answers
16 views

Limit of solution equal to solution at limit

I have a system of equations (not DE, think algebraic equations) that depends on a parameter. I am trying to learn under what conditions on the system is a limit (as the parameter converges to some ...
2
votes
1answer
41 views

Finding solutions to a system of linear equations

I was working on this problem, to which I have the answer, but it is just that, an answer with no explanation, and I am stuck on how the answer was arrived at, for a few parts of this question. In ...
0
votes
1answer
20 views

Apply gauss method to a linear system and them use results in another system

I have an exercise for my last assignment of linear algebra, which is the following: I tried to row reduce to echelon form the matrix created by the first linear system of equations and I obtain ...
0
votes
2answers
18 views

Don't understand adding a system of compound inequalities

I'm reading a proof of the Division Theorem and one line that comes up is Since 0 ≤ r1 < b and 0 ≤ r2 < b , we have −b < r1 − r2 < b. I do not ...
1
vote
3answers
179 views

Find $p$ for which all solutions of system/equation are real

There is system of $5$ equations $$ a+b+c+d+e = p; \\ a^2+b^2+c^2+d^2+e^2 = p; \\ a^3+b^3+c^3+d^3+e^3 = p; \\ a^4+b^4+c^4+d^4+e^4 = p; \\ a^5+b^5+c^5+d^5+e^5 = p, \\ \tag{1} $$ where $p\in\mathbb{R}$. ...
0
votes
1answer
23 views

simultaneous equations-exponential and linear

I am trying to find a general formula for x and y given that $y=mx+c$ and $y=Ae^{kx}$, with m, c, A and k as constants (and e is Euler's number). essencially, find the point(s) where an exponential ...
0
votes
1answer
23 views

System of equations with 3 variables - Does order matter?

I'm studying Algebra and I'm now at topic 'System of equations with 3 variables'. I'm having a hard time with the following example: $$ \begin{cases} 2x + 2y + 3z = 10\\ 3x + y-z = 0\\ x + ...
1
vote
0answers
45 views

Nonnegative solution of a linear system

Given three collections of parameters $\epsilon_1 > ... > \epsilon_N$, $(a_1,...,a_{N-1})$ and $(b_1,...,b_N)$ that satisfy the following conditions: (i) $\forall i, a_i \geq 0, ...
0
votes
1answer
29 views

How to solve over-determined linear system of equations?

at the moment I am working on an application where I have to solve some systems of linear equations during the whole algorithm. Because the programming-language I have to use is something related to ...
0
votes
2answers
116 views

System of $24$ variables

Assume that $a_1, a_2,\ldots, a_{24}$ satisfy $$a_1+a_2+\ldots+a_{24}=26$$$$a_1^2+a_2^2+\ldots+a_{24}^2=26$$$$\vdots$$$$a_1^{24}+a_2^{24}+\ldots+a_{24}^{24}=26$$ Find $a_1a_2⋯a_{24}$. How do I solve ...
0
votes
0answers
36 views

Shifting RGB colors based on ratio

I have two colors represented in RGB (foreground(fr, fg, fb) and background(br, bg, bb). Based on a relative luminance value, a color difference value and a brightness value i'd like to figure out the ...
1
vote
0answers
36 views

Differentiating both sides of a DE

In general if you have a differential equation with two variables such that: $$L(x,y)=h_1[f(x),f'(x),f^{(2)}(x),...,f^{(n)}(x),g(y),g'(y),g^{(2)}(y),...,g^{(n)}(y)]\\ ...
0
votes
0answers
14 views

Distribution of unknown, given system of equations

Suppose we have an unknown real $x$. We want to give an approximation of $x$ by measuring the distance between $s_i$ and $x$, for $i = 1,2,3$. The position of each $s_i$ is distributed with mean $p_i$ ...
0
votes
1answer
21 views

Logarithmic equation in which base of the logarithm is not same.

If $log_{9}x=log_{12}y=log_{16}(x+y)$ then find $y/x$. I simplified this into $\frac { \log { x } }{ \log { 9 } } =\frac { \log { y } }{ \log { 12 } } =\frac { log(x+y) }{ \log { 16 } } $ I ...
2
votes
3answers
121 views

Solve this system of equation

Solve this system of equations for real $x$ and $y$: $5x\left(1+\dfrac{1}{x^2+y^2}\right)=12$ $5y\left(1-\dfrac{1}{x^2+y^2}\right)= 4$ I juggled with those equations and got ...
2
votes
2answers
36 views

Conditional Extremum, need help finding the extreme points in calculation.

Find the conditonal extremums of the following $$u=xyz$$ if $$(1) x^2+y^2+z^2=1,x+y+z=0.$$ First i made the Lagrange function $\phi= xyz+ \lambda(x^2+y^2+z^2-1) + \mu (x+y+z) $, now making the ...
-1
votes
2answers
19 views

To test following system of linear equation for equivalency

Let F be field of complex numbers I have two system of equations $x_1 - x_2 =0 $ $2x_1 + x_2 =0$ And $3x_1 + x_2 -0$ $x_1 + x_2 =0$ The definition says that each if equation in first system ...
0
votes
1answer
18 views

The phrase “periodic boundary conditions” for a two-variable PDE

I'm currently working on trying to solve a system of PDE's of the form $c_t=D_x(c_{xx} + c_{yy})+K_1 c + K_2 d$ $d_t= D_y(d_{xx}+d_{yy})+K_3 c + K_4 d$ that has "periodic boundary conditions" on a ...
0
votes
0answers
28 views

Numeric solution for a non-linear system

It has been a while I have not practiced mathematics but I should have enough background to get your answers if well detailed. I have an n-by-n matrix, let's call it D, where dij represents the ...
0
votes
1answer
26 views

Advice to solve a system of 8th order univariate polynomials

I am struggling to solve a least square problem in which the tedious part is the initialization. Grid search methods are out of question. The initial problem I've stated my problem in a previous ...
0
votes
1answer
30 views

Find the solution to the system (not linear)

Find all $(x, y, z) \in \mathbb{R^3}$ satisfying: $$x^2 + 4y^2 = 4xz \tag1$$ $$y^2 + 4z^2 = 4xy \tag2$$ $$z^2 + 4x^2 = 4yz \tag3$$ This is a very difficult problem. I added $-4(1) + (3)$ to ...
2
votes
1answer
19 views

system of equations with $n$ equations and $2^k n$ unknowns

I have a system of equations with infinitely many solutions. I would like to find a "nice" way to write down an explicit solution. Here, $n,k\geq 1$ are integers, we have $x_1,x_2,\dots, x_{2^k n}$ ...
0
votes
2answers
29 views

Explain how solution got $c_1$ and $c_2$

Can someone explain how the solution manual got $c_1$ and $c_2$ in this:
0
votes
0answers
20 views

Solving a system of differential-algebraic equations

I am seeking for references around the topic of solving this type of differential equation system : $$ \left\{ \begin{array}{ll} \partial_x y_{i+1}(x) = y_{i+1}(x)-y_{i}(x) \\ y_{i+1}(x) = y_i(f(x)) ...
1
vote
0answers
28 views

Convert a 2D autonomous ODE system into a 1D system?

Suppose I have two equations: $$\frac{dx}{dt} = x(2-x-y),\, \frac{dy}{dt} = ky(2-ax-by)$$ that together form a 2D ODE system ($x$ plotted against $y). K, a$ and $b$ are all independent positive ...
0
votes
0answers
26 views

Conormal Points Parabola

Let the line $lx+my=1$ cut the parabola at $y^2=4ax$ in the points A and B.Normals at A and B meet at a point C. Normal from C other than these two meet at D.Then coordinates of D are? I tried to ...
0
votes
0answers
28 views

Systems of equations of the form $\sum_{i \in I} \sum_{j \in J_i} v_i \times v_j = a$

Is there any theory that deals (directly or not) with systems of equations of the form $$\sum_{i \in I} \sum_{j \in J_i} v_i \times v_j = a,$$ where $a \in \mathbb{R}^3$ is known, $v_i, v_j \in ...
0
votes
1answer
33 views

Optimizing for the minimum relative distance in a given situation?

I have primarily been working on this problem for quite some time now; the level of the problem is introductory calculus w/ optimization problems. The situation is as follows: Ship A sails due ...
0
votes
0answers
21 views

Ordinary differential equation system with maximum value of x+y

I've got the following simple system of ODEs: x': x*(1-x-y) y': y*(1-a* y-b* x). Plotting the system in a x' y' plot is not fully satisfying to me as what I want to achieve is that the sum of x and ...
5
votes
2answers
88 views

How to prove whether the equation set has a unique solution?

\begin{eqnarray} \begin{cases} \sin A \sin C-(\sin B)^2=0 \cr AC-B^2=0 \cr A+B+C-\pi=0 \cr A>0,B>0,C>0 \end{cases} \end{eqnarray} How to prove whether the equation set has a unique solution ...
0
votes
3answers
23 views

multiplying term on sum

Say I know the following relation holds $$ \sum_i f_i + \sum_i g_i = 0 $$ Now I multipy both sides with a set of vectors $\mathbf v_i$. Will it still be true that $$ \sum_i f_i \mathbf v_i + \sum_i ...
1
vote
2answers
33 views

Solving Systems of Linear Differential Equations by Elimination

For a homework problem, we are provided: $\frac{dx}{dt}=-y + t$ $\frac{dy}{dt}=x-t$ Putting these into differential operator notation and separating the dependent variables from the independent: ...
1
vote
0answers
19 views

I need help with creating linear equations from multiple points on a graph

How do you create a linear equation from multiple points on a graph I am working on a question where the points are $(-3,8),(2,5)$ and $(7,2)$ and I need to find out how to create a linear inequality ...
1
vote
1answer
25 views

solution of a system of equation

Let $A\in M_{m\times n}(\Bbb R)$ and let $b_0\in \mathbb R^m$. Suppose that the system of equations $Ax=b_0$ has a unique solution. Which of the following is true? $Ax=b$ has a solution for every $b ...
1
vote
2answers
120 views

Solve a system of two nonlinear equations

$$ \begin{cases} x^2 - y^2 + 12y - 21 = 0\\ 2x^2 + y^2 + 2xy + x = 0 \end{cases} $$ I've tried the change of variables: $u = x + y$, $v = x - y$ After it I've got: $$ \begin{cases} uv + 12\frac{u - ...
2
votes
1answer
46 views

A system of linear equations with 3 variables such that its solution set is: $\{(a,b,c)|a^2=b \}$?

Is there a system of linear equations with 3 variables such that its solution set is: $\{(a,b,c)|a^2=b \}$? It's enough to show that for one equation: $Ax+By+Cz=D$ the solution set doesn't work. ...
1
vote
2answers
35 views

Solve the following in vector form:

So i did a substitution to solve the system normally, and got $x=17.67$ $y=9.67$ $z=10.67$ Where I am stuck is how to represent something like this in a vector form, maybe my solution was wrong ...
1
vote
1answer
39 views

Rearranging An Equation To Solve?Can't?

How would I rearrange the equation: $$a=b^{(c/d)}$$ to find c?
1
vote
0answers
37 views

least square solution of overdetermined system with additional unknown

I was hoping somebody could tell me the best way to solve the following overdetermined system for the scalars $x_{1}$,$x_{2}$ and $x_{3}$, where the C $3 \times 1$ vectors are unknown, $A_{i}$ is a $3 ...
1
vote
0answers
23 views

canonical form of parabolic-type PDE involving exp(x) and ln(x)

The attached picture skips a lot of the work, but I've worked this problem at least 6 times in the last 8 hours, still getting stuck at reducing to canonical form - that is, trying to solve for x and ...
0
votes
2answers
31 views

3 Variable System of Equations When All Set to Zero

So I'm doing an a bit of a pre-assessment for something, and I feel like I am missing something on this question: Now I know how to solve a normal 3 variable system, but with this they are all set ...
1
vote
1answer
49 views

Analytic solution of a system of four second order polynomials

Can I systematically solve in $\mathbb{R}^4$ the following system without using Grobner basis algorithm ? If not, can I find the exact number of solutions ? $$ \begin{equation*} \left\{ ...
0
votes
1answer
26 views

Solution for associated homogeneous linear system

If an inhomogeneous system of linear equations has an associated homogeneous system that has only the trivial solution, then how can I show that the inhomogeneous system has exactly one solution?
6
votes
1answer
152 views

Amount of solutions to the Diophantine equation of Frobenius

The Diophantine equation of Frobenius is any equation of the form: $$\sum_{i=1}^k a_i x_i = n$$ where the $a_i$'s are given and so are $k$ and $n$. I'm looking for an algorithm to compute the number ...
0
votes
1answer
58 views

Is it possible to create any combination of areas?

Given a point $P(x,y)$ in the unit square, two polygons Blue and Green are defined by drawing a 45-degree line through $P$ and creating polygons with the top-left and bottom-right corners, ...
1
vote
1answer
26 views

Behaviour of roots of a polynomial with function coefficients

Let $(-1+c_4(h))x^4 +c_3(h)x^3+c_2(h)x^2+c_1(h)x+c_0(h)=0$ be an equation with variable coefficients, depending smoothly on $h$. Also let $0\le c_4(h)\le 1-\epsilon$ for some $\epsilon>0$ and ...