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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1answer
61 views

Ratio & simultaneous linear equation

A pharmacist needs to combine a $2\%$ solution of a medication with a $25\%$ solution (of the SAME medication) to make $9$ litres of a $3\%$ solution. Use simultaneous linear equations to determine ...
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6answers
40 views

How to find the area of a triangle with two equations?

So I was given the following problem : ABC is a right angled triangle with the sides $a,b,c$ . Find the area of this triangle, given that $$a+b+c = 22$$ $$a^2+b^2+c^2 = 200$$ I've tried to do a lot ...
1
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1answer
22 views

Is there a general formula for the $n$'th variable of the solution for a lower triangular linear system of equations?

I have a countably infinite linear system of equations $Ax = b$, where $A$ is lower triangular with $-1$ at all diagonal entries, and $b = \{-1/2,0,0,...,0\}^T$. I.e the $n$'th unknown depends solely ...
2
votes
2answers
158 views

Solving Three equations for 3 Unknowns

Today I have a question and I am really curious to know about this. Question: $$ 16y+39z+50zy=0$$ $$ 85x-78z+95zx=0$$ $$ 85x+32y+70xy=0$$ $$\text{Are The Equations like these can be solve for ...
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1answer
35 views

Proof by induction, system of equations

We conjecture that there is a formula of the form $\sum_{j=1}^{n}{j^2} = an^3 + bn^2 + cn + d$ for all integers n ≥ 1 (3) (a) Assuming that such a formula is true, find the value of a, b, c, d. (...
0
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0answers
15 views

4x5 linear equation treaded as parameter

I got a 4x5 linear equation (4 equation 5 incognitas)like this: 1 1 0 0 0 = 800 0 1-1 1 0 = 300 0 0 0 1 1 = 500 1 0 0 0 1 = 600 i tried to give solution taking ...
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1answer
23 views

Have do i get 2nd equation from 3 equation 2 variable system answer?

My teacher today solved this system of equations for us that consisted of these 3 equations, 1) p0 + p1 =1 2) a*p0 + b*p1 =p0 3) c*p0 + d*p1 =p1 , ...
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3answers
25 views

Find coefficient in linear equation system for non-trivial solution (easy question)

Find the real value of $\alpha$ so that the system admits solution different from (0, 0, 0). $\begin{cases} \alpha x + y = 0 \\ \alpha y + z = 0 \\ 8x + \alpha z = 0 \end{cases}$ a) 8 b) 2 c) 1 d) -...
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0answers
43 views

How to solve these simultaneous equations?

I'm doing questions from this page: http://tartarus.org/gareth/maths/tripos/IB/Variational_Principles.pdf and I'm doing Question 2013 1/I/6A The question asks to find the cylindrical cup of least ...
0
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1answer
18 views

Convert line parametrization into two equations

Consider the following parametrization on $\mathbb{R}^3$ $$g(t) = (t^2,t\cos(t),t\sin(t))$$ This is a line, and as such can be characterized by two equations. I already found the first one to be $$...
2
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1answer
27 views

Properties of the solution of a linear system with random equations

$x_i$ is drawn from $\mathrm{unif}(a,b)$, $y_i$ is drawn from $\mathrm{unif}(c,d)$. $x_i$ are independent from each other. $y_i$ are independent from each other. $x_i$ are independent of $y_i$. $i$ ...
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votes
2answers
66 views

Basis for intersecting subspaces - is there a trick here?

I'm doing this problem, which gives me these subspaces of $\mathbb{R}^4$ $$U=\text{span}\left\{\;\begin{pmatrix} 3\\ 2\\4 \\ -1\end{pmatrix},\;\begin{pmatrix} 1\\ 2\\1 \\ -2\end{pmatrix},\;\begin{...
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votes
2answers
50 views

Solution to system of non linear equations

what is the best way to solve this system of equations: $$ax^2 +by^2-2y=0$$ $$axy+byz-z=0$$ $$ay^2+bz^2-c=0$$ Solve for x,y,z where a,b,c are constants.
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0answers
85 views

System of Equations which can be solved by inequalities: $(x^3+y^3)(y^3+z^3)(z^3+x^3)=8$, $\frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}=\frac32$.

S367. Solve in positive real numbers the system of equations: \begin{gather*} (x^3+y^3)(y^3+z^3)(z^3+x^3)=8,\\ \frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}=\frac32. \end{gather*} Proposed by ...
0
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2answers
61 views

Determine $x$ if $x = 4 \mod 17$ and $x = 3 \mod 11$. [closed]

Given $x =4\mod 17$ and $x = 3\mod 11$, determine $x$. I know that $\gcd(17,11)= 1$. I was hoping to use this to determine $x$.
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1answer
27 views

If $A$ is a singular square matrix, then $Ax = b \neq \vec{0}$ has $0$ or many solutions

I was reading this pdf: https://www.math.ohiou.edu/courses/math3600/lecture10.pdf and it tells you that if $A$ is a singular square matrix, then $Ax = b \neq \vec{0}$ has $0$ or many ...
1
vote
1answer
35 views

How can I write a set of equations in summation form?

I have a system of equations as follows: \begin{align} & A_1^{11} + A_1^{12} + A_1^{13} + \cdots + A_1^{1n}=X \\[8pt] & A_1^{21} + A_1^{22} + A_1^{23} + \cdots+ A_1^{2n}=X \\[8pt] & \...
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0answers
18 views

Counting the number of roots of multivariate polynomials?

The equation of a circle is well known $$(x-x_0)^2+(y-y_0)^2 - r^2 = 0$$ It has a solution all along the circle with midpoint $(x,y) = (x_0,y_0)$. We also know that $ab = 0$ whenever any of $a$ and/or ...
0
votes
1answer
38 views

solving a pair of simultaneous equations

I have a rather messy pair of simultaneous equations, which I need to solve for x: $\left(x+2n-1\over2\right)^2+\left(\sqrt{1-\left(x^2-2\over2\right)^2}+\sqrt{1-\left(-x^2+x+2n+1\over2\right)^2}\...
1
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2answers
45 views

So many logs with different bases

$ \large { 6 }^{ \log _{ 5 }{ x } }\log _{ 3 }( { x }^{ 5 } ) -{ 5 }^{ \log _{ 6 }{ 6x } }\log _{ 3 }{ \frac { x }{ 3 } } ={ 6 }^{ \log _{ 5 }{ 5x } }-{ 5 }^{ \log _{ 6 }{ x } }$ The sum of the ...
1
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2answers
47 views

Number of integral solutions for an equation

How do we approach this kind of problem of finding number of positive integral solutions to $$\frac{1}{x}+\frac{1}{y} = \frac{1}{n!}$$ Here $n$ is given.
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votes
1answer
10 views

How is Dulac's Multiplier selected?

I'm aware that Dulac's (Negative) Criterion states that a system of differential equations of the form $x' = f(x,y), \; y' = g(x,y)$ has no periodic orbits in the plane if we can find some function $\...
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0answers
21 views

Finding the General Solution for a System of Differential Equations with Complex Eigenvalues

I think I might just be having trouble with formatting my answer, because I'm fairly sure my work is right up until this point. The question asks to find the general solution to $$X'= \begin{bmatrix}...
0
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0answers
19 views

Euler explicit and semi-implicit

I am given a simple dynamic system with an initial condition: $a(t) = 0.9 - 0.1v(t)$ $v(0) = x(0) = 0$ I want to calculate $x(1)$ with a time step of $\Delta t = 1$ using Euler explicit and semi ...
0
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0answers
26 views

Nonlinear equations algorithm - Newton method

Some time ago I posted a question regarding the simple case of finding the intersection point when I have only two functions, and with your help I found an answer. It was this case: $f(x) = a + \...
2
votes
1answer
76 views

A seemingly-trivial divisibility conjecture

While working on another problem, I stumbled on the following divisibility claim. Conjecture: No integers $a,b,c,d$ satisfy all of the following conditions: $a^2+b^2-c^2-d^2 = 2(ad-bc)-1$; $\gcd(ac+...
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2answers
33 views

Solving system of non-linear equations.

So I'm trying to find the stationary points for $$f(x,y,z) = 4x^2 + y^2 +2z^2 -8xyz$$ Setting the partial derivatives to zero leads to: $$x-yz=0 \\ y-4xz=0\\z-2xy=0$$ Substiting $z=2xy$ into the ...
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3answers
40 views

Real problems solved with systems

Can anybody tell me where can I find some REAL problems (i.e. form real life) that can be solved using a 3x3 system of linear equations? Or, can anybody give me an example? A solution could be a ...
1
vote
1answer
29 views

Nonlinear equations systems

Can anybody help me to find a system with 3 equations and 3 unknowns and a bounded domain D = [a,b]x[c,d]x[e,f] such that the system has an unique solution in D? Also, i need nice equations, because ...
3
votes
2answers
50 views

The condition about some positive real numbers can be written as the sum the nearby two

Given $n$ positive real numbers $x_1,...,x_n$. What is the condition that they can be written as $$x_1=y_1+y_2$$ $$x_2=y_2+y_3$$ $$\ldots$$ $$x_n=y_n+y_1$$ where $y_1,\ldots,y_n$ are also some ...
0
votes
1answer
71 views

System of equation $x+y+z=2007; xyz=14000$

I have to solve the system of equations $$\begin{cases} x+y+z=2008,\\ xyz=14000, \end{cases}$$ where $x,y,z$ are positive integers such that $1\le x \le y \le z \le 2000.$ My work so far: Let $...
0
votes
1answer
27 views

Consistency of system of linear equations

Find when the equations $$\begin{cases}x + y - 2z = 0\\ax + by + cz = 0\\bx + cy + az = d\end{cases}$$ are consistent and solve them completely when they are consistent. I have tried the ...
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votes
0answers
48 views

Solution of Nonlinear System of Equations

I wand to find the solutions $p_H, p_L$ implied by the following two equations: (I) $$\frac{(1-\lambda)(p_L-c_L)}{p_H \frac{q_L}{q_H}-c_L} = \frac{\lambda(p_L-c_L)}{p_H-(q_H-q_L)-c_L} - \lambda$$ (...
0
votes
0answers
25 views

How to Solve this Nonlinear System of Equations?

There are 24 variables and 24 equations in the system: $ i=0,1,2,3\\ \textrm{variables}: s_i,\ t_i,\ a_m \; (m=0,...,15)\\ \textrm{constants}: b_{ni} \; (n=0,...,5)\\$ $$\begin{array}{rcl} a_0\cdot ...
2
votes
6answers
126 views

Solve the system of equations: $a+b+c=2$, $a^2+b^2+c^2=6$, $a^3+b^3+c^3=8$ [closed]

If we have \begin{cases} a+b+c=2 \\ a^2+b^2+c^2=6 \\ a^3+b^3+c^3=8\end{cases} then what is the value of $a,b,c$?
1
vote
2answers
73 views

The sum of two numbers is 5/9…

The sum of two numbers is $\frac{5}{9}$. The quotient of the two numbers is $1$. What is the product of $40$% of each number? The answer I got was $\frac{1}{81}$. I don't understand this - would ...
0
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0answers
32 views

What are the eigenvalues of the following Hermitian matrix?

Let $\mathtt{i}=\sqrt{-1}$ and $$p=1+\mathtt{i}=\bar{q},\ \ q=1-\mathtt{i}=\bar{p}.$$ Let $A$ be an $n\times n$ matrix such that $$A=\begin{bmatrix} 0 & p & p & \cdots & p & \...
0
votes
1answer
22 views

how do you solve the simultaneous equations 2x + y = 9 and x - 2y = -8 using the matrix method?

i first convert the bases into a 2x2 matrix and then i multiplied the inverse matrix by the 2x1 e/f matrix of 9 (on top) and -8 (on the bottom) which gave me x = 5.4 and y = 5 however the answer ...
0
votes
1answer
26 views

Find the length of the longest diagonal of the bo

The total length of all $12$ sides of a rectangular box is $60$. The surface area of the box is given to be $56$. Find $(i)$ the length of the longest diagonal of the box $(ii)$ the volume of the box ...
2
votes
1answer
58 views

Symmetric system of equations problem

Solve the following simultaneous eqations on the set of real numbers: $$a^2+b^3=a+1$$ $$b^2+a^3=b+1$$ I have found two trivial solutions: $$a=b=1$$ $$a=b=-1$$ but I can't prove that there are no ...
0
votes
0answers
23 views

How to numerically minimize system of equations composed of data and smoothness terms, ensuring minimum solution norm

I need to find $g$ that minimizes: $$\sum_{v=0}^n (f+g_{v_{left}}-g_{v_{right}})^2 + \frac{1}{\lambda}\sum_{v=0}^m (g_{v_i}-g_{v_j})^2$$ where $f$ is constant and the sums are over pair of $v$ indices;...
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0answers
25 views

Reducing a system to first order

Convert the following to a first order system $$x''(t) = k_x(x(t) - y(t))^{-2}, \ \ y''(t) = k_y(x(t) - y(t))^{-2},$$ $$x'(0)=v_x, \ \ y'(0) = v_y, \ \ x(0) = x_0, \ \ y(0) = y_0.$$ I know how to ...
0
votes
0answers
8 views

Closed-form solution for system of equations for finding a critical point

I am trying to find a critical point of a function $\mathbb{R}^d \to \mathbb{R}$ by setting its gradient to zero. I would like to solve the follwoing system of equations. $$\frac{1}{1 - \sum_{j=1}^d ...
1
vote
1answer
39 views

Systems of equation

Find non-negative solutions of systems of equations: $$\begin{cases} x^2y^2+1=x^2+xy \\ y^2z^2+1=y^2+yz \\ z^2x^2+1=z^2+zx \end{cases} $$ My work so far: 1) $(1;1;1) - $ solution. 2) $(y^2-1)x^2-...
1
vote
3answers
72 views

What does x equivalent to 2 mod 15 mean?

I came across the following question: Consider the following system of equivalences of integers. $$ x \equiv 2 \bmod{15} $$ $$ x \equiv 4 \bmod{21} $$ The number of solutions in $x$, where $1\le x\...
1
vote
1answer
40 views

Solve $A^kx=b$ system using $LU$

I have the system $A^kx=b$ and the $LU$ factorization $A=LU$. How can I solve the system without actually calculating $A^k$?
1
vote
1answer
103 views

Solve this systems to condition $3x^3(x+1)^2=2y^2(z+3)^3$

if $x,y,z$be postive real numbers, solve systems of this following equation $$ 3x^3(x+1)^2=2y^2(z+3)^3\tag{1}$$ $$3y^3(y+2)^2=2z^2(x+1)^3\tag{2}$$ $$3z^3(z+3)^2=2x^2(y+2)^3\tag{3}$$ My approach is ...
4
votes
2answers
65 views

Why is a polynomial $f(x)$ sum of squares if $f(x)>0 $ for all real values of $x$?

If a polynomial $f(x)>0 $ for all real values of $x$, then $f(x)$ is sum of squares. Why is this true ? I understand that the roots of this $f(x)$ will be complex and hence will exist as ...
0
votes
0answers
15 views

Sum of triangular matrices system

I was wondering if there is a nice way to solve the following linear system of equaitons: $(A+B) x = b$, where $A$ is an upper-right triangular matrix (all elements higher than the main diagonal are ...
-1
votes
1answer
27 views

Rewriting system as a set of first order equations.

What I'm given: $$x'' = x' + y' + x + y$$ $$y'' = 2x' + 3y' + 3x + y$$ $$z=x'$$ $$w=y'$$ My solution: We know that $z'=x''$ and $w'=y''$. We can write: $$z'=z+w+x+y$$ $$w'=2z+3w+3x+y$$ I'm not ...