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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21
votes
5answers
703 views

Solving a peculiar system of equations

I have the following system of equations where the $m$'s are known but $a, b, c, x, y, z$ are unknown. How does one go about solving this system? All the usual linear algebra tricks I know don't apply ...
19
votes
9answers
6k views

System of nonlinear equations that leads to cubic equation

The system of equations are: $$\begin{align}2x + 3y &= 6 + 5x\\x^2 - 2y^2 - (3x/4y) + 6xy &= 60\end{align}$$ I can solve it through substitution but it is an arduous process to reach this ...
17
votes
8answers
3k views

Kid's homework: 4 equations 5 unknowns? Going crazy!

I'm new here, and I'm hoping someone can help out. My 10 year old son has been set a maths problem, which I can't solve. I've got a PhD in neuroscience and do a fair amount of matlab stuff (data ...
14
votes
4answers
764 views

How find the value of the $x+y$

Question: let $x,y\in \Bbb R $, and such $$\begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases}$$ Find the $x+y$ This problem is from china some BBS My idea: since ...
12
votes
2answers
668 views

System of non-linear equations.

I have to find all triplets $(x,y,z)$ that satisfy: $$x^{2012} + y^{2012} + z^{2012} = 3\\x^{2013} + y^{2013} + z^{2013} = 3\\x^{2014} + y^{2014} + z^{2014} = 3$$ I've found the trivial solution ...
11
votes
4answers
565 views

If $2^x=3^y=6^{-z}$ then prove that:$ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0$

If $$2^x=3^y=6^{-z}$$ and $x,y,z \neq 0 $ then prove that:$$ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0$$ I have tried starting with taking logartithms, but that gives just some more equations. Any ...
11
votes
1answer
166 views

Cyclic system of equations

Consider the system of equations $$ \begin{align*} x^2+(1-y)^2&=a\\ y^2+(1-z)^2&=b\\ z^2+(1-x)^2&=c\\ \end{align*} $$ Compute $x(1-x)$ in terms of $a,b,c$. Edit: The question should say ...
10
votes
6answers
1k views

what would be the way to solve a system of equations like this one?

Solve: $xy=-30$ $x+y=13$ {15, -2} is a particular solution, but, how would I know if is the only solution, or what would be the way to solve this without "guessing" ?
10
votes
3answers
1k views

A system of equations with 5 variables

Find the real numbers $a, b, c, d, e$ in $[-2, 2]$ that simultaneously satify the following relations: $$a+b+c+d+e=0$$ $$a^3+b^3+c^3+d^3+e^3=0$$ $$a^5+b^5+c^5+d^5+e^5=10$$ I suppose that the key is ...
9
votes
2answers
389 views

Solving for unknown functions

I am not a mathematician, so excuse if my question is silly or badly stated. I have the following problem. I have 2 conditions on two unknown continuously differentiable functions: ...
8
votes
5answers
281 views

Given $x+y$ and $x\cdot y$, what is $x^3+ y^3$ ?

I have been looking at an assortment of high school number sense tests and I noticed a reoccurring problem that states what x+y is and what $x\cdot y$ is then asks for $x^3+ y^3$. I want to know how ...
8
votes
1answer
73 views

Solving $x_1+x_2=x_3^2, x_2+x_3=x_4^2, x_3+x_4=x_5^2,x_4+x_5=x_1^2, x_5+x_1=x_2^2$ in reals

find answers of this system of equations in real numbers$$ \left\{ \begin{array}{c} x_1+x_2=x_3^2 \\ x_2+x_3=x_4^2 \\ x_3+x_4=x_5^2 \\ x_4+x_5=x_1^2 \\ x_5+x_1=x_2^2 \end{array} \right. ...
8
votes
1answer
125 views

Method of characteristics for systems of PDE (vs. Lewy's example)

Main question: How does the method of characteristics generalize for systems of first order PDE, as opposed to scalar PDE? Namely, is there such a generalization at all, and if so what information ...
7
votes
1answer
361 views

The solutions for the equation $\frac{a}{c-b+1}+\frac{b}{a-c+1}+\frac{c}{b-a+1}=0.$

How can I find the solution for the following equation in $a,b \mbox{ and } c$. $$\frac{a}{c-b+1}+\frac{b}{a-c+1}+\frac{c}{b-a+1}=0.$$ Also $b-c \neq 1$, $c-a \neq 1$ and $a-b \neq 1$. Thanks!
7
votes
4answers
356 views

How to solve the equations $\sqrt{x-3}+\sqrt{y-3}=\sqrt{y-12}+\sqrt{z-12}=\sqrt{z-27}+\sqrt{x-27}=12$

Let $x,y,z\in R$, and $$\begin{cases} \sqrt{x-3}+\sqrt{y-3}=12\\ \sqrt{y-12}+\sqrt{z-12}=12\\ \sqrt{z-27}+\sqrt{x-27}=12 \end{cases}$$ Find the $x,y,z$. My try: I want use The geometry to ...
7
votes
3answers
292 views

Finding the all integer solutions

How to Find the all integer solutions for: $$x+y+z=3$$ $$x^3+y^3+z^3=3$$
7
votes
6answers
378 views

Solve $5a^2 - 4ab - b^2 + 9 = 0$, $ - 21a^2 - 10ab + 40a - b^2 + 8b - 12 = 0$

Solve $\left\{\begin{matrix} 5a^2 - 4ab - b^2 + 9 = 0\\ - 21a^2 - 10ab + 40a - b^2 + 8b - 12 = 0. \end{matrix}\right.$ I know that we can use quadratic equation twice, but then we'll get some ...
7
votes
2answers
218 views

How to solve this system for real $x,y,z$

Find the real values $x,y,z$ such that $$\begin{cases} x+y^2+z^3=21\qquad (1)\\ y+z^2+x^3=71\qquad (2)\\ z+x^2+y^3=45\qquad (3) \end{cases}$$ Thank you everyone. This problem have some nice methods, ...
7
votes
1answer
117 views

How do we solve this system of equations?

$a,b \in \Bbb R$ and $$\frac{a^5b-b^5a}{a-b}=30$$ and $$a^5+b^5 = 33$$ I get that $a^6-b^6=(a-b)63$ But I have no idea how to solve after that. Someone could help me?
7
votes
2answers
119 views

Very simple partial differential equation

I am solving $$ \frac {\partial f}{\partial x} = \frac y{x^2 + y^2} \\ \frac {\partial f}{\partial y} = \frac {-x}{x^2 +y^2} $$ As $y$ was held constant when the partial derivative with respect to ...
7
votes
1answer
192 views

infinite matrix leading eigenvalue problem

I'm trying to find the leading eigenvalue and corresponding left and right eigenvectors of the following infinite matrix, for $\lambda>0$: $$ \mathrm{A}=\left( \begin{array}{cccccc} 1 ...
6
votes
3answers
728 views

Three-variable system of simultaneous equations

$x + y + z = 4$ $x^2 + y^2 + z^2 = 4$ $x^3 + y^3 + z^3 = 4$ Any ideas on how to solve for $(x,y,z)$ satisfying the three simultaneous equations, provided there can be both real and complex ...
6
votes
3answers
269 views

How to solve this infinite set of equations?

If I can find a solution to the following set of equations then, with a bit of luck, I should be able to derive all sorts of nifty new results in non-equilibrium statistical mechanics. However, I'm ...
6
votes
2answers
291 views

Transition time in a Lotka-Volterra system

I am working with a set of real-valued ordinary differential equations based on the Lotka-Volterra competition equations: $$\begin{align} \dot{a_1} & = a_1 \left( 1 - a_1 - 2 a_2 \right) \\ ...
6
votes
3answers
238 views

Integer Programming problem

I have an integer programming problem with $L$ variables $x_1, x_2, x_{L}$ which all assume integer values and the following constraints must stand: $x_i \geq 0$ $x_1 = 10$ $x_2 + x_3 + ... + x_{L} ...
6
votes
2answers
186 views

System of two Equations

A friend of Mine gave me a system of two equations and asked me to solve them $\rightarrow$ $$\sqrt{x}+y=11~~ ...1$$ $$\sqrt{y}+x=7~~ ...2$$ I tried to solve them manually and got this horrendously ...
6
votes
4answers
94 views

Solve the System of Equations in Real $x$,$y$ and $z$

Solve for $x$,$y$ and $z$ $\in $ $\mathbb{R}$ if $$\begin{align} x^2+x-1=y \\ y^2+y-1=z\\ z^2+z-1=x \end{align}$$ My Try: if $x=y=z$ then the two triplets $(1,1,1)$ and ...
6
votes
2answers
121 views

Stability analysis for ODEs with non constant inputs

For a project, I have to deal with systems of ODE's with non constant input such as: $$\begin{cases}\dot x =I(t)x+x^2\\ \dot y=x\end{cases}$$ where I(t) is a random input (for example). In any case, ...
6
votes
0answers
124 views

How to solve a particular initial-boundary value problem

I have the following initial-boundary value problem $$\begin{cases}\dfrac{\partial^2 u_1}{\partial x^2}=A_{11}\dfrac{\partial u_1}{\partial t}+A_{12}\dfrac{\partial u_2}{\partial ...
5
votes
4answers
885 views

A Five Equations problem?

If, $$\begin{align*} y+u+x+v&=0\\ z+y+v+u&=1\\ x+y+z+u&=5\\ z+u+v+x&=2\\ v+x+y+z&=4\,, \end{align*}$$ What is the value of $xyzuv$?
5
votes
4answers
536 views

What is the non-trivial, general solution of these equal ratios? [closed]

Provide non-trivial solution of the following: $$\frac{a}{b+c}=\frac{b}{c+a}=\frac{c}{a+b}$$ $a=?, b=?, c=?$ The solution should be general.
5
votes
2answers
219 views

How to solve these three equations?

If α ,β ,γ are three numbers s.t.: $\ α^ \ $ + $\ β \ $ + $ γ \ $ = −2 $\ α^2 \ $ + $\ β^2 \ $ + $ γ^2 \ $ = 6 $\ α^3 \ $ + $\ β^3 \ $ + $ γ^3 \ $ = −5, then $\ α^4 \ $ + $\ β^4 \ $ + $ ...
5
votes
2answers
2k views

System of equations: $x^2+y=7, y^2+x=11$ [duplicate]

Possible Duplicate: Steps to solve this system of equations During the flight from Moscow to Yerevan my neighbor gave me the following problem: Solve the system: ...
5
votes
3answers
339 views

Solving a simple system of equations

Given the simultaneous equations $$A\cos{(\sqrt{\lambda}\pi)} + B\sin{(\sqrt{\lambda}\pi)} = 0$$ $$A\cos{(2\sqrt{\lambda}\pi)}+B\sin{(2\sqrt{\lambda}\pi)} = 0$$ We want to show this has not trivial ...
5
votes
5answers
837 views

Solving a set of recurrence relations

I have the 7 following reccurence relations: $A_n = B_{n-1} + C_{n-1}$ $B_n = A_n + C_{n-1}$ $C_n = B_n + C_{n-1}$ $D_n = E_{n-1} + G_{n-1}$ $E_n = D_n + F_{n-1}$ $F_n = G_n + C_n$ $G_n = E_n + ...
5
votes
2answers
561 views

Solve a linear system with more variables than equations

Suppose that, after a series of elementary row operations the augmented matrix of a linear system with variables $x_1$, $x_2$, $x_3$, $x_4$ is transformed into reduced row echelon form as follows: ...
5
votes
4answers
529 views

Linear Algebra - four “true or false” questions about matrices and linear systems

I'm reviewing for my linear algebra course, and have four "true or false" questions that I'm struggling to prove. I've included my approach to the solutions in brackets below them: 1) If $A^2 = B^2$, ...
5
votes
2answers
212 views

Can it be decidable for any polynomials to have the intersecting point?

Give system of polynomials$$P_1(x_1,x_2,\dots,x_n)=0,$$$$\vdots,$$$$P_k(x_1,x_2,\dots,x_n)=0$$ Can it be decidable for thoses polynomials to have the intersecting point ?
5
votes
2answers
128 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 ...
5
votes
2answers
238 views

Solved a quartic equation by dividing

I was asked to solve: $$x^4+2x^3-22x^2+2x+1 = 0$$ Without using differential calculus (Newton's Method). My Progress: Dividing by $x^2$, I get: $$x^2+2x-22+\frac{2}{x}+\frac{1}{x^2} = 0$$ $$x^2 ...
5
votes
2answers
254 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 ...
5
votes
2answers
96 views

Solve $x+3y=4y^3,y+3z=4z^3 ,z+3x=4x^3$ in reals

Find answers of this system of equations in reals$$ \left\{ \begin{array}{c} x+3y=4y^3 \\ y+3z=4z^3 \\ z+3x=4x^3 \end{array} \right. $$ Things O have done: summing these 3 equations give ...
5
votes
2answers
210 views

Seemingly simple system of equations

I have the following system: $x^{2} + y = 31$ $x + y^{2} = 41$ As I try to solve it via simple substitution, I get into 4-th power equations, which I can simplify to $(x-5)(x^{3}+5x^{2}-37x-184)$ ...
5
votes
1answer
134 views

When is the equation $Ax = b$ solvable in the integers?

Let $A$ be an $m\times n$ matrix with integer entries, $b$ a column-vector with $m$ integer entries. Suppose the equation $Ax = b$ has infinitely many solutions. It is clear that the general ...
5
votes
2answers
132 views

Why does the following nonlinear system have 21 solutions?

I am curious as to why the following nonlinear system has 21 solutions (according to Wolfram Alpha). $$y+xy^2-x^3+2xz^4=0 \\ -x-y^3-3x^2y+3yz^4=0 \\ -\frac{5}{2}y^2z^3-2x^2z^3-\frac{z^7}{2}=0$$ I ...
5
votes
2answers
72 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 ...
5
votes
1answer
82 views

For a system of PDEs, how many equations are needed generally for the system to have unique solution?

For an algebraic system of equations or a system of ordinary differential equations the following rule holds:(right?) the total number of unknown variables must be equal to the number of equations ...
5
votes
0answers
144 views

System of 3 equations

I am doing thermal calculation in electronics and when trying to device a general formula for equivalent system resistance to air flow of a part of real system, I ended with this system of three ...
5
votes
2answers
82 views

System of non-linear ODE's

do you have any suggestions to solve analytically the Non-linear ODE system $\dot x=18 x^2 y-3p x^2+6p xy$ $\dot y=18 x^2 y-6p xy $ where $p$ is a real constant. Thank you very much cheers
4
votes
6answers
648 views

Find the value of $x$ and $y$ given this equation

So I have a College Admission test tomorrow and I am hoping that you could help me understand how to arrive at the solution to this: 1.) Given the following equations: $$3x-y=30\\ 5x-3y=10$$ ...