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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29
votes
5answers
2k views

System of 4 tedious nonlinear equations: $ (a+k)(b+k)(c+k)(d+k) = $ constant for $1 \le k \le 4$

It is given that $$(a+1)(b+1)(c+1)(d+1)=15$$$$(a+2)(b+2)(c+2)(d+2)=45$$$$(a+3)(b+3)(c+3)(d+3)=133$$$$(a+4)(b+4)(c+4)(d+4)=339$$ How do I find the value of $(a+5)(b+5)(c+5)(d+5)$. I could think only of ...
22
votes
5answers
832 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
806 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 ...
13
votes
4answers
3k views

Proof that any linear system cannot have exactly 2 solutions.

How would you go about proving that for any system of linear equations (whether all are homogenous or not) can only have either (if this is true): One solution Infinitely many solutions No solutions ...
13
votes
4answers
368 views

Solve the system $ x \lfloor y \rfloor = 7 $ and $ y \lfloor x \rfloor = 8 $.

Solve the following system for $ x,y \in \mathbb{R} $: \begin{align} x \lfloor y \rfloor & = 7, \\ y \lfloor x \rfloor & = 8. \end{align} It could be reducing to one variable, but it is ...
12
votes
2answers
694 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 ...
12
votes
4answers
950 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 ...
12
votes
4answers
918 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\\ ...
11
votes
3answers
2k 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 ...
10
votes
2answers
385 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 ...
9
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" ?
9
votes
5answers
317 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 ...
9
votes
5answers
868 views

How to solve an exponential and logarithmic system of equations?

$$ \left\{\begin{array}{c} e^{2x} + e^y = 800 \\ 3\ln(x) + \ln(y) = 5 \end{array}\right.$$ I understand how to solve system of equations, logarithmic rules, and the fact that $\ln(e^x) = e^{\ln(x)} ...
9
votes
2answers
648 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: ...
9
votes
1answer
93 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. ...
9
votes
2answers
157 views

How to prove the cubic formula without root extraction

I'm trying to prove the cubic formula, in the following form: Given a field $F$ and $x,p,q\in F$, define $m=\frac p3$ and $n=\frac q2$, and suppose also that $\gamma,\tau$ are given such that ...
9
votes
1answer
318 views

How prove this systems-equation has least two postive integers solution

Show that: for any $k\ge 100,(k\in N^{+})$, there exsit $p\in N^{+}$, such $$\begin{cases} a+b+c=k\\ abc=p\\ a>b>c \end{cases}$$ has at least two postive integers solution $(a,b,c)$ ...
9
votes
1answer
394 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 ...
8
votes
1answer
139 views

Evaluate $a^2+b^2+c^2$

I found this questions from past year maths competition in my country, I've tried any possible way to find it, but it is just way too hard. If $a, b, c$ are distinct numbers such that $a^2 - bc = ...
8
votes
4answers
193 views

Is there a simpler approach to these system of equations?

I recently came across the following system of equations: $$x + y + z = 1 \\ x^2 + y^2 + z^2 = 2 \\ x^3 + y ^3 + z^3 = 3$$ And I have two questions: One, is there a way to prove or disprove ...
8
votes
2answers
277 views

Prove or Disprove the Existence of Solutions…

Let $A$ be a $3\times 4$ and $b$ be a $3\times 1$ matrix with integer entries.Suppose that the system $Ax=b$ has a complex solution. Then which of the following are true? (CSIR December 2014) ...
8
votes
2answers
198 views

Pentagonal Numbers

I recently was passing some time on Project Euler, when I came across this question. It deals with finding Pentagonal Numbers $P_j$ and $P_k$ such that $P_j+P_k$ and $P_j-P_k$ are also pentagonal ...
8
votes
1answer
104 views

Solve: $x = (x-\frac{1}{x}) ^ {1/9} + (1-\frac{1}{x})^{1/9}$

Solve: $$x = \left(x-\frac{1}{x}\right) ^ {1/9} + \left(1-\frac{1}{x}\right)^{1/9}$$ Simplifying, $$x^{10/9} = (x^2-1)^{1/9}+(x-1)^{1/9}$$ I don't know how to start. Any hint will be helpful.
7
votes
7answers
1k views

Finding x in an Olympiad simultaneous equation

I have been practicing for a an upcoming intermediate math olympiad and I came across the following question: Let $x$ and $y$ be positive integers that satisfy the equations $$\begin{cases} xy = ...
7
votes
1answer
377 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
636 views

Exponential Simultaneous Equations

Solve the following simultaneous equations: $$2^x + 2^y = 10$$ $$x + y = 4$$ Looking at it, it is obvious that the answers are $(3,1)$ and $(1,3)$, however, I was wondering if they could be solved ...
7
votes
2answers
2k 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: ...
7
votes
4answers
452 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
311 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
2answers
146 views

How many pairs of positive integers (x,y) that satisfy equation $ x^2 - 10! = y^2$?

I have a question about number theory. How many pairs of positive integers $(x,y)$ that satisfy equation $$x^2 - 10! = y^2$$ ? My attempt: Move the $y^2$ from right to the left and 10! From left to ...
7
votes
2answers
233 views

Solve $xy=3$ and $4^{x^2}+2^{y^2}=72$

I have a system of equations $xy=3$ and $4^{x^2}+2^{y^2}=72$ whose solution I know is $x=y=\sqrt 3$, but what are the steps to solve it?
7
votes
6answers
413 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
242 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
2answers
329 views

System $a+b+c=4$, $a^2+b^2+c^2=8$. find all possible values for $c$.

$$a+b+c=4$$$$a^2+b^2+c^2=8$$ I'm not sure if my solution is good, since I don't have answers for this problem. Any directions, comments and/or corrections would be appreciated. It's obvious that ...
7
votes
4answers
77 views

Prove that for any given $c_1,c_2,c_3\in \mathbb{Z}$,the equations set has integral solution.

$$ \left\{ \begin{aligned} c_1 & = a_2b_3-b_2a_3 \\ c_2 & = a_3b_1-b_3a_1 \\ c_3 & = a_1b_2-b_1a_2 \end{aligned} \right. $$ $c_1,c_2,c_3\in \mathbb{Z}$ is given,prove that $\exists ...
7
votes
1answer
122 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
133 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
238 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 ...
7
votes
2answers
158 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
5answers
1k views

Some equations from Russian maths book. [closed]

Could you please help me with solving these equations. I would like to solve them in the most sneaky way. All of the exercises in this book can be solved in some clever way which I can't often find. ...
6
votes
6answers
820 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: $$3x-y=30\\ 5x-3y=10$$ ...
6
votes
4answers
503 views

How to prove that the following system of equations has only one solution?

$ \begin{cases} (x - 1)^2 + (y + 1)^2 = 25 \\ (x + 5)^2 + (y + 9)^2 = 25 \\ y = -\frac{3}{4}x - \frac{13}{2} \end{cases} $ I have to solve this system of equations. After substituting $y = ...
6
votes
3answers
996 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
297 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
6answers
129 views

System of equations involving sin and cos

I'm trying to solve the following system: $$ \sin(x) + \cos(y) = 0.6\\ \cos(x) - \sin(y) = 0.2\\ $$ Solving for y in terms of x: $$ y=\arccos(0.6-\sin(x))=\arcsin(\cos(x) -0.2) $$ Therefore: $$ ...
6
votes
2answers
106 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 ...
6
votes
2answers
345 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
268 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} ...