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
105 views

Solving a system of five polynomials

I am trying to solve the following system of equations for tuple $\left(a,b,c,d,t\right) \in \mathbb{R}^{4} \times [0,1]$, with parameter $\ell\in\mathbb{R}$. $$ \begin{eqnarray} a\frac{t^{2}}{2} - ...
1
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0answers
34 views

A question about a system of PDE

It is well known that under suitable conditions, the symmetry of mixed second partial derivatives reads: $$\frac{\partial^2 f}{\partial x \partial y}=\frac{\partial^2 f}{\partial y \partial x}.$$ ...
2
votes
1answer
37 views

A general method for solving systems of quadratic equations

For linear systems we have general methods (i.e. Gauss elimination). Is there a general method for solving systems of quadratic equations with many variables? I heard about Groebner bases; is there ...
0
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1answer
24 views

consistency of solution question

Let $A, B$ be $n\times n$ matrices and $c, d$ be $n \times 1$ vectors such that the matrix equations $$Ax = c$$ $$Bx = d$$ are consistent, i.e., each equation admits a solution. Can we conclude that ...
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3answers
73 views

$ x+y = 1 $ and $ \frac{1}{x} + \frac{1}{y} = 1 $ Solve $ x^3 + y^3 $ [closed]

$x$, $y$ are complex numbers, $x$ and $y$ aren't $0$. $$ x + y = 1 $$ $$ \frac{1}{x} + \frac{1}{y} = 1 $$ $$ x^3 + y^3 = ? $$ Thank You!
4
votes
1answer
75 views

Another troubling system of equations

I've been working on solving some linear equations arising from different optimization problems, but I keep getting stuck. Right now I have the problem below: I am trying to solve the system of ...
3
votes
5answers
77 views

Using equation to find value of $1/x - 1/y$

$$\left(\frac{48}{10}\right)^x=\left(\frac{8}{10}\right)^y=1000$$ What is the value of $\frac{1}{x}-\frac{1}{y}$? I have already used that when $48$ divided by $10$ then it becomes $4.8$ and when $8$ ...
0
votes
0answers
19 views

What is the best time complexity for this case?

I only want to know if the following system has any integer solution or not. Actually, I do not need to know the solution(s), and only need to know the answer of question "Does the system have any ...
0
votes
0answers
17 views

Set of 3 inequations involving 3 unknowns with a maximum

I am capable of finding a relation between unknowns x, y and z involved in this set of 3 inequations: $\begin{cases} ax - y - z \leq x \\ -x + by - z \leq y \\ - x - y + cz \leq z\end{cases}$ This ...
1
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0answers
24 views

Given a set of arbitrary data, is it possible to model this data using differential functions.

Problem At the moment, I have a problem with seven variables: $S, A_1, A_2, R_1, R_2, P_0, P_1 $ and $P_2$. Each of these variables draws a smooth line through time. My question is, is there any ...
4
votes
3answers
133 views

How to solve a system of logarithmic equations?

I need to create a function with the following properties: $$f(1)=1$$ $$f(65)=75$$ $$f(100)=100$$ Additionally, the function needs to grow logarithmically. So that gives three equations: $$A \cdot ...
2
votes
0answers
21 views

$L$-existential and $L$-diophantine

Could you explain to me the last sentence: "Whenever we want to stress dependence on the language, we will use the self-explanatory terms and $L$-existential and $L$-diophantine" ? What does ...
1
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1answer
50 views

What does the code do and what ODE is that?

This exercise I came across asks what kind of method of solving ODE is that: ...
1
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1answer
36 views

Equation for spacing of elements on the edge of a circle

I'm trying to come up with an equation which, given an index within an arbitary number of elements (the most natural example would be 12, as in 12 numbers on a clock), along with an arbitrary radius, ...
3
votes
2answers
66 views

Solve $x+y+z=1; x^2+y^2+z^2=35; x^3+y^3+z^3=97$

It may be surprising that I can't get any analytical way of verifying that one of the solutions of $$x+y+z=1$$ $$x^2+y^2+z^2=35$$ $$x^3+y^3+z^3=97$$ is $x=-1, y=-3$ and $z=5$. Although it may be ...
1
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0answers
57 views

system of non-homogeneous advection equations

I would like to solve this system \begin{equation} \left\{ \begin{array}{lll} u_t+b_1 u_x=(r+l_1)u-l_1v,\\ v_t+b_2 v_x=(r+l_2)v-l_2u,\\ \end{array} \right. \end{equation} First , I would like to ...
0
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1answer
22 views

Solving a system of equations with an absolute value term

$x$ and $y$ are two integer numbers and $x \geq y$. The values of $x$ and $y$ are positive or negative integers. When the sum of these two numbers are multiplied by $y$ we obtain $P$ and when the ...
0
votes
0answers
39 views

How to find a formula for ratios?

I don't know if this is the correct section to post this, but here it goes. I recently got involved with hydroponics, and to feed the plants I've installed a system with a pump that delivers a ...
0
votes
0answers
130 views

Second order coupled differential equations

I am trying to solve two coupled ordinary differential equation. $x''+Ax'+By'+Cx+Dy=U$ ; $y''+Ex'+Fy'+Gx+Hy=V$ ; $A,B,C,D,E,F,G$ & $H$ are constants, $U,V,x$ and $y$ are function of time. All ...
2
votes
2answers
72 views

Solution to system of linear ODE's

Let $\Delta_n$ be the closed unit simplex in $\mathbb R^n$. For any $a,b \in \Delta_n$, define the differential equation: $$ a'(u) = b-a(u) \quad\quad\quad a(0) = a $$ How does one go about solving ...
1
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1answer
26 views

System of Linear Equations 2 variables

A theater sold 160 children’s tickets and 90 adult tickets. If the theater made $1,600 from the sales of the tickets, what were the prices of each ticket? My set up is: ...
1
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1answer
55 views

Mapping the intersection of hyperplanes/simplex to lower-dimensional unit-simplex

Suppose I have an object in $\mathbb{R}^5$ described by: $$x_1+x_2+x_3+x_4+x_5=1$$ $$x_1+2x_2+3x_3+4x_4+5x_5=6$$ $$x_1+7x_2+8x_3+9x_4+10x_5=11$$ $$x_1,x_2,x_3,x_4,x_5 \geq 0$$ Is there a way that I ...
1
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1answer
22 views

Given two variables and their ranges, get a third value.

I'm building a model but I got stuck at this: I have $x,y$ whose ranges are ($10,000$ to $2,000,000$) and (1 to 36) respectively. Also I have a z that ranges from 16 to 30. I know their relations in ...
1
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1answer
30 views

Find the end points of a line segment in 3D space

I have a line segment in 3 dimensional space (x,y,z), and I want to find the 2 endpoints of this line segment. Is there a systematic way of doing this? To be specific, I have the line described by ...
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votes
2answers
63 views

how to solve nonlinear system of equations

I have $A, B, C, D, E, F.$ I want to calculate a and b from the following system of equations: I know I should solve this system using $3$ equations and $3$ unknowns, but it is not linear. can any ...
2
votes
1answer
31 views

System of Congruences with Special Symmetry

Show that the following system of congruences \begin{align} \begin{cases} 3 x^4 - 7 x^2 y^2 - 7 x^2 z^2 - 35 y^2 z^2 \equiv 0 \pmod{p} \\ 3 y^4 - 7 x^2 y^2 - 7 y^2 z^2 - 35 x^2 z^2 \equiv 0 \pmod{p} ...
3
votes
0answers
41 views

Solving simultaneous equations in complex numbers

Given $z_1,z_2$ are complex numbers such that sum of their squares is a real number and $$z_1(z_1^2-3z_2^2)=2$$ and $$z_2(3z_1^2-z_2^2)=11.$$ I need to find the value of sum of squares of two complex ...
0
votes
1answer
9 views

Is there a general relationship between number of variables and of constraints?

This comes up most notably in linear algebra and differential equations - usually unique solutions come about when the number of constraints matches the number of variables, or in the case of ...
0
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0answers
38 views

How to solve complicated systems of equations with exponentials?

I'm trying to solve the system of equations at the bottom of this image, I need to solve for Rohm, Rdf, Rct, Cdl, Cdf, and Voc in terms of the theta parameters so I can calculate them from the ...
0
votes
2answers
32 views

Determining the coordinates of a vector with respect to a basis

Problem: Let $V = \mathbb{R}[X]_{\leq 4}$ be the vectorspace of all polynomials of degree at most $n$, and let $\alpha = \left\{1, 1 +x, (1+x)^2, (1+x)^3, (1+x)^4\right\}$ be a basis for $V$. Find the ...
3
votes
1answer
49 views

Intuitive Explanation For Why Dependent Equations Contain No Added Information?

I've always been taught that because dependent equations contain no added information they can be deleted without effecting the solution set. Now this makes sense to me if an equation is a constant ...
3
votes
1answer
63 views

transform integral to differential equations

I found a similar system of integral equations in a paper. It says that it can be solved by differentiating and then using standard techniques. My question is, how can I differentiate such a system in ...
0
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3answers
239 views

Find the value of $x$ such that $\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-x}}}}=x$

Find the value of $x$, $$\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-x}}}}=x$$ Help guys please, I have tried and I got, $x=-2, x=1$, and I think it's wrong
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votes
2answers
58 views

For what values of $a$, $b$, and $c$ the above system has: One solution. Infinitely many solutions. No solutions.

I am stuck with this now, I tried reducing the matrix to row echelon form, but it gets a bit hard. Is there not a simpler way? The system is: \begin{align*} a x + b y − 3 z &= −3\\ −2 x − b y + ...
1
vote
3answers
57 views

Simultaneous equation with fractional solutions.

How do you get to find $x$ when $y$ is a fraction ? Anyone mind to explain it step by step for the clearest explanation.=) $$-7x +2y = 2$$ $$14x + 3y = -5$$ Answer: $x=?, y=-1/7$
1
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1answer
341 views

Apollonian gasket

Okay , is there a way to find the radius of the nth circle in a apollonian gasket .. Something like this Its like simple case of apollonian gasket .. I found from descartes' theorem $R_n = ...
0
votes
1answer
118 views

A good equation system

Given $a,b,c$ positive numbers, solve the system $\sqrt{xy}+\sqrt{xz}-x=a$, $\sqrt{yz}+\sqrt{yx}-y=b$ and $\sqrt{zx}+\sqrt{zy}-z=c$, where $x,y,z\in \mathbb{R}$. This only a pretty question. I did ...
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votes
3answers
286 views

If 6x = y+z and 4x = y-z, express z in terms of x

\begin{align} 6x &= y+z\\ 4x &= y-z \end{align} How to express $z$ in terms of $x$? I'm not 100% sure on how to solve in terms of x
1
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4answers
48 views

How to find a real matrix with complex eigenvalues,

Give a $2 \times 2$ real matrix $A$ with eigenvalues $2+3i$, $2-3i$. I would like hints only. So far, I've been trying get somewhere with $\det[A-(2+3i)I] = 0$ and $\det[A-(2-3i)I] = 0$; which ...
11
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4answers
622 views

Solving a system of non-linear equations

Let $$(\star)\begin{cases} \begin{vmatrix} x&y\\ z&x\\ \end{vmatrix}=1, \\ \begin{vmatrix} y&z\\ x&y\\ \end{vmatrix}=2, \\ \begin{vmatrix} z&x\\ y&z\\ ...
5
votes
1answer
168 views

Equivalence of system of nonlinear equations

Let $A\in\mathbb{R}^{n\times n}$ be a positive semidefinite matrix, $b\in\mathbb{R}^n$, $k>0$, and $g:\mathbb{R}^n\rightarrow\mathbb{R}$ be a positive function. Consider the system of nonlinear ...
0
votes
0answers
25 views

Can I solve these simultaneous questions, and if so, how?

I have a set of paired experimental observations (Fo, Bo), e.g. 1.55, 8.52 4.56, 36.36 21.03, 64.98 (> 6 data pairs in total) which I believe can be modelled as being generated by Fo = Ft + a.Bt ...
5
votes
2answers
59 views

System of equations in a,b,c,d

$a,b,c,d$ are complex numbers satisfying \begin{cases} a+b+c+d=3 \\ a^2+ b^2+ c^2+ d^2=5 \\ a^3+ b^3+ c^3+ d^3=3 \\ a^4+ b^4+ c^4+ d^4=9 \end{cases} Find the value of the following: ...
0
votes
0answers
15 views

Another trigonometric moment problem

Is there a standard approach for solving the following system: $$ m_k = \sum_{j=1}^N a_j e^{-2\pi i \mu_j k \delta}, \quad k = 0, 1, 2, \ldots, $$ where $N \in \mathbb{N}$, $m_k \in \mathbb{C}$, ...
5
votes
4answers
627 views

Why are the coefficients always equal?

Take the equation $ax^{2} + bx + c = 3x^{2} + 4x + 53$. Why is it always true that $a = 3, b = 4$ and $c = 53$? I've seen many examples like this where the coefficients are equated, and was just ...
1
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1answer
41 views

Underdetermined vs Overdetermined Problem

I'm trying to create a model which is of the form $$y = (a_0 + a_1l)[b_0+\sum_{m=1}^M b_m\cos(mx-\alpha_m)] [c_0 +\sum_{n=1}^N c_n\cos(nz-\beta_n)]$$ In the above system, $l$,$x$ and $z$ are ...
1
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2answers
32 views

Grasping “Substitution” in terms of linear algebra

So I have a set of equations: $$x_{1} + x_{2} = 1$$ $$x_{2} + x_{4} = 3$$ From linear algebra, we know that (say, we're in $\mathbb{R}^{4}$, i.e. we have 4 variables), the solution space to the ...
1
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0answers
23 views

How can I solve this specific set of equations?

Here are the equations: $$\sum_{k = 1}^n i_k + Y_n u_n = J \quad \quad (1)$$ $$i_1 + Y(u_1 - u_2) = J \quad \quad (2)$$ $$i_k - Y(u_{k - 1} -2u_{k} + u_{k + 1}) = 0, \quad \quad k = 2, ..., n - 2 ...
3
votes
2answers
60 views

How to solve this system of equations for $x^2+y^2+z^2$?

For the complex numbers $x,y,z$, the system of equations $x^2-yz=i~~~~~ y^2-zx=i~~~~~ z^2-xy=i$ It is not easy for me to get $x^2+y^2+z^2$ from the above. I don't need the values of $x,y,z$ I'm ...
1
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3answers
71 views

When do variables cancel out?

Sometimes if I randomly combine different equation and try to solve for a variable, one of them will cancel out. Why? For example: $\displaystyle x^2 = 4y^2$ and $\displaystyle x = 2y + 1$ And solve ...