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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5
votes
5answers
3k views

Steps to solve this system of equations: $\sqrt{x}+y=7$, $\sqrt{y}+x=11$

I want to solve this system of equations, I have been out of Maths for a long!! $$\sqrt{x}+y=7$$ $$\sqrt{y}+x=11$$ Just wondering easiest step to find values for $x$ and $y$ from the above ...
6
votes
3answers
1k 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 ...
5
votes
2answers
3k 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: $$\left\{\begin{array}{c}x^...
-1
votes
7answers
376 views

Systems of linear equations to calculate $\alpha$ and $\beta$

Point $1$: When there is $1$ car passing the road, the average speed is $50$ km/h. Point $2$: When there are $5$ cars passing the road, the average speed is $45$ km/h. Point $3$: When there are $12$ ...
8
votes
2answers
370 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) $...
7
votes
2answers
19k views

Help with using the Runge-Kutta 4th order method on a system of 2 first order ODE's.

The original ODE I had was $$ \frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0$$ with $y(0)=3$ and $y'(0)=1$. Now I can solve this by hand and obtain that $y(1) = 14.82789927$. However I wish to use the 4th order ...
9
votes
5answers
330 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 ...
2
votes
2answers
133 views

Solve system of 3 equations

$x+y+z=0$ $x^2+y^2+z^2=6ab$ $x^3+y^3+z^3=3(a^3+b^3)$ this is what i reasoned out so far; $xyz=a^3+b^3$ $x^2+zx+z^2=3ab$ $y^2+zy+z^2=3ab$ $x^2+xy+y^2=3ab$ $y^2=3ab+zx$ $x^2=3ab+zy$ $z^2=3ab+...
1
vote
3answers
575 views

System of Linear Equations - how many solutions?

For which real values of t does the following system of linear equations: $$ \left\{ \begin{array}{c} tx_1 + x_2 + x_3 = 1 \\ x_1 + tx_2 + x_3 = 1 \\ x_1 + x_2 + tx_3 = 1 \end{array} \right. $$ ...
19
votes
2answers
535 views

Does this system of simultaneous Pell-like equations have any non-trivial positive integer solutions?

Let $a,b,c$ be positive integers satisfying \begin{align} 2a^2-1 &= b^2, \\ 2a^2+1 &= 3c^2. \end{align} The trivial solution is $(a,b,c)=(1,1,1)$. Are there others?
22
votes
5answers
873 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 ...
13
votes
4answers
4k 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 ...
4
votes
1answer
7k views

Polar coordinates differential equation

I have the following ODE: $$\dot x=-y(x^2+y^2), \dot y=x(x^2+y^2)$$ I want to sketch the phase portrait (manually) and I want to find the flow $\phi_t$, the orbit $O(x_0)$ and the limit set $\omega(...
3
votes
4answers
119 views

Quick way to solve the system $\displaystyle \left( \frac{3}{2} \right)^{x-y} - \left( \frac{2}{3} \right)^{x-y} = \frac{65}{36}$, $xy-x+y=118$.

Consider the system $$\begin{aligned} \left( \frac{3}{2} \right)^{x-y} - \left( \frac{2}{3} \right)^{x-y} & = \frac{65}{36}, \\ xy -x +y & = 118. \end{aligned}$$ I have solved it by ...
5
votes
4answers
600 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.
1
vote
2answers
210 views

$10$ Equations in $10$ variables

Define, $$F_k = xp^k + yq^k + zr^k + us^k + vt^k$$ Let, $$F_0 = 2$$ $$F_1 = 3$$ $$F_2 = 16$$ $$F_3 = 31$$ $$F_4 = 103$$ $$F_5 = 235$$ $$F_6 = 674$$ $$F_7 = \color{blue}{1667}$$ $$F_8 = ...
4
votes
2answers
84 views

How to solve simultaneous inequalities?

I am doing multivariable calculus, and specifically double integrals. I am facing difficulties finding the domain of the integal, however i am given the following equations: $$1 ≤ 2x+y ≤ 2$$ $$0 ≤ x-...
2
votes
1answer
56 views

Existence of solution of $\frac{\partial f}{\partial t}=-\Delta f+|\nabla f|^2-R(x,t)$

When $t=t_0$, $f(x,t)=f_0(x)\in L^2(U)$. $t\in [0,t_0]$ and $U$ is a open subset of $R^n$.$R(x,t)$ is bounded and smooth about $x$ and $t$. I don't whether suitable the conditions is ,if not, please ...
1
vote
3answers
8k views

How to plot a phase portrait for this system of differential equations?

I beg your help.. I'd like the phase portrait for this system. I don't know how to use Mathematica/Matlab ... :( If anyone can make this portrait and post a print screen here, I would thank you ...
5
votes
1answer
188 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 ...
4
votes
1answer
178 views

Find basis of solutions of this linear system.

I am supposed to find basis of the subspace of vector space $ \mathbb{R}^{3} $ of solutions of this linear system of equations: $y = \left\{ \begin{array}{ll} x_{1}+2x_{2}-x_{3}=0 \\ 2x_{1}+7x_{2}-2x_{...
4
votes
2answers
161 views

A calculus problem with functions such that $f''(x) = g(x)$ and $g''(x) = f(x)$

Let: $f(x)$ and $g(x)$ be twice differentiable, non-decreasing functions. $f''(x) = g(x)$ and $g''(x) = f(x)$. $f(x) \cdot g(x)$ is a linear function. Then we have to show that $f(x) = g(x) = 0$....
3
votes
1answer
60 views

system of matrix equations

I have the equation $ \mathbf{x}^T A \mathbf{x} = b $, where $b$ is a scalar, $\mathbf{x}$ a vector of size $M$, and A a matrix of size $M\times M$. $b$ and $\mathbf{x}$ are given. How many such ...
2
votes
1answer
72 views

Specific system of differential equations

I have the following system of equations: \begin{eqnarray}\frac{dx}{dt} = x(1 - x^2 - y^2) \\ \frac{dy}{dt} = y(4 - x^2 - y^2) \end{eqnarray} I want to prove that if a solutions starts (at time $t = ...
2
votes
1answer
297 views

Eigenvectors Trajectories

I got stuck with a problem while studying for a control systems exam. It goes as following: "Look at the picture of trajectories of a linear, time-invariant system with the form: $\frac{d\mathbf{x}}{...
2
votes
5answers
158 views

Why a linear numerator for fractions with irreducible denominators?

For example: (2x^3+5x+1)/((x^2+4)(x^2+x+2)) breaks down to (ax+b/(x^2+4))+(cx+d/(x^2+x+2)). I have been told that since the denominators are irreducible, the numerators will be either linear or ...
1
vote
2answers
60 views

How to solve for any given natural number n?

I started with hard way of putting the coefficients into a matrix. But, iz did not help. the following system of linear equations: system
1
vote
1answer
166 views

Solve equation with unknown in exponents

This is in continuation of this but not related to it completely. I am interested in finding a solution to the equation: $m' = m - \sum \limits_{j=1}^{m} (1 - d_{O_j}/n)^k$. where $m,m',n$ and $d_{...
1
vote
1answer
70 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
2answers
68 views

trouble with non-homogeneous ODE system… which method shall I use?

I am an undergrad statistics student and I am having troubles with non-homogeneous ODE systems. During my classes I went over just three methods for solving odes: Laplace transform, Fourier transform ...
0
votes
1answer
81 views

Linear systems. Please help me solve this

Please help me solve this. Consider for every real number $a$ the linear system of equations: $$ \begin{align} x +( a + 1 )y + a^2 z &= a^3 \\ (1-a)x +( 1 - 2a )y &= a^3 \\ x +( a +...
-2
votes
3answers
872 views

Solution of 3 equations in 3 unknowns [closed]

Find the value of $c$ which makes it possible to solve: $$u+v+2w=2,$$ $$2u+3v-w=5,$$ $$3u+4v+w=c$$
-2
votes
0answers
68 views

Writing systems of linear equations and matlab help appreciated [duplicate]

Does anyone know how I would go about answering the following question? A traffic engineering company has installed some trac cameras along a one lane road to find a relation between the number of ...
-2
votes
0answers
62 views

Writing systems of linear equations [duplicate]

Would anyone be able to help me with the following question? Thank you. Point 1: When there is 1 car passing the road, the average speed is 50km/h. Point 2: When there are 5 cars passing the road, ...
14
votes
4answers
811 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 $$(3x^3+4y^3)(4x^4+3y^...
8
votes
4answers
204 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 ...
0
votes
2answers
117 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 ...
5
votes
2answers
280 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
5answers
2k 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 + ...
4
votes
0answers
163 views

When do two integral superellipses have 'nice' intersections?

A recent question posed the nonlinear system \begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases} for real $(x,y)$ and asked for the sum $x+y$. As noted by commentary in the question, this regrettably ...
3
votes
3answers
277 views

Show that: a) $X^{-1}(t)$ is bounded in $[\beta,\infty)$. b)No system solution approaches zero solution when $t \rightarrow \infty.$

Let a system $x' = A(t)x$ and suppose there are values positives $k, \beta$ such that a positive fundamental matrix $X(t)$ satisfies $\|X(t)\| \leq k$, $t \geq \beta$ and $$ \liminf_{t \rightarrow \...
3
votes
2answers
4k views

Find the value of k, (if any), for which the system below has unique, infinite or no solution. [duplicate]

The system of equations are: $\begin{cases}x+y+kz = 1\\x+ky+z=1\\kx+y+z=1\\ \end{cases}$ I am looking to finding values of $k$, for which this system has either no solutions, infinite many solutions ...
2
votes
1answer
586 views

Method of characteristics for a system of pdes

I can do parts a) and b) as follows $\begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&1\end{pmatrix}\frac{\partial}{\partial{}x}\begin{pmatrix} u \\ v \\ w\end{pmatrix}+\begin{pmatrix} 1&...
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 $a,b,c$?
2
votes
3answers
162 views

How was the determinant of matrices generalized for matrices bigger than $2 \times 2$?

How was the determinant of matrices generalized for matrices bigger than $2 \times 2$? I read a book a very long time ago where it said something like this: Given a system of two equations with two ...
0
votes
1answer
117 views

Solutions to simultaneous Diophantine equations $2y^2-3x^2=-1$ and $z^2-2y^2= -1$

I am looking for integer solutions for the following set of equations: $2y^2-3x^2=-1$ $z^2-2y^2= -1$ I know that there are the solutions (1,1,1) and (-1,-1,-1) for this set of simultaneous ...
6
votes
4answers
116 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 $(-1,-1,-...
5
votes
2answers
99 views

General solution to a system of non linear equations with a specific pattern

I am seeking a general solution to a system of non linear equations with a specific pattern: Order 1: $$ x_0 = a^2 + b^2 $$ $$ x_1 = 2ab $$ Order 2: $$ x_0 = a^2 + b^2 + c^2 $$ $$ x_1 = 2ab + 2bc $...
5
votes
8answers
69k views

Finding two numbers given their sum and their product

Which two numbers when added together yield $16$, and when multiplied together yield $55$. I know the $x$ and $y$ are $5$ and $11$ but I wanted to see if I could algebraically solve it, and found I ...
5
votes
2answers
3k views

Sum of cubed roots

I need to calculate the sums $$x_1^3 + x_2^3 + x_3^3$$ and $$x_1^4 + x_2^4 + x_3^4$$ where $x_1, x_2, x_3$ are the roots of $$x^3+2x^2+3x+4=0$$ using Viete's formulas. I know that $x_1^2+...