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
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2answers
71 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 ...
0
votes
2answers
60 views

How do you find the value of n in this example

$$n^{n-2} = 16$$ I know $n = 4$ through trial and error but how do you find $n$ in a conventional manner? I'm basically trying to solve how many nodes are in a tree that has $16$ spanning trees ...
2
votes
1answer
8 views

Equation for sinusoidal wave with fixed wavelength and amplitude

I am a programmer. I am writing a program in which I need to show a graph plotted to the user when the user adjusts two sliders, which are the amplitude and wavelength of the wave, say...
4
votes
1answer
83 views

Knight movement on chess field

I had this task in programming competition: There are two knights, which are $(p_1,q_1)$ and $(p_2, q_2)$. $(p,q)$ knight is figure, with p(q)-length first step, and q(p)-length second step in ...
1
vote
1answer
34 views

Weird contradiction between equations

A guy that I tutor came to me with the following question: The time it takes for body $A$ to pass 160 km is 5 hours longer than the time it takes for body B to pass 90 km. The speed of body A is ...
6
votes
4answers
72 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 ...
1
vote
2answers
32 views

Writing system of equations & rate of change

"Two planes leave a city for another city that us 600 miles away. One of the planes is flying 50 miles per hour faster than the other. The slower plane takes 2 hours longer to reach the city. What is ...
1
vote
1answer
33 views

Prove that the solutions to the system of equations are integers

Let $a, b \in \mathbb{Z}$ and consider the system of equations below: $$\begin{cases} y -2x-a =0\\ y^2-xy+x^2-b=0\end{cases} $$ Prove that $x,y\in\mathbb{Q}$ implies $x,y\in\mathbb{Z}$. I ...
2
votes
3answers
76 views

Matrix Problem of form Ax=B

The matrix $A$ is given by $$\left(\begin{array}{ccc} 1 & 2 & 3 & 4\\ 3 & 8 & 11 & 8\\ 1 & 3 & 4 & \lambda\\ \lambda & 5 & 7 & 6\end{array} \right)$$ ...
1
vote
1answer
44 views

Relatively simple system of nonlinear ODEs

There are a lot of questions like this on MSE as well as online resources on the subject, but a) the MSE questions are either unanswered or correspond to systems substantially different from this one, ...
2
votes
1answer
19 views

System of DEs with constant term

This is similar but not identical to standard examples in e.g. Paul's Notes, and while the math seems straightforward the results I get disagree with what I get from numerical simulation. Given a 2D ...
0
votes
0answers
17 views

show that $y$ and $x$ can be expressed as a function of $x$ alone in the neighbourhood of the point $(2,1,-1)$

Let $$f(x,y,z)=[x^2-2xz+y^2z^3-7,2xy^4-3y^2+xz^2+5z+2]=(0,0)$$ at $$(2,1,-1)$$ The only method that I have known is the one for solving the homogenous system say for $y$ and $z$ in terms of $x$ but ...
0
votes
2answers
28 views

Existence if integer solutions

How many integer solutions exist for: \begin{cases} x+y=1-z \\ x^3+y^3=1-z^2 \end{cases} How do I do this. I'm stuck
1
vote
3answers
45 views

Consider the parametric curve: $x=6\cos^3(t), y=6\sin^3(t)$, write it in cartesian form.

Consider the parametric curve: $$x=6\cos^3(t), y=6\sin^3(t)$$ Write it in Cartesian form. I am really struggling with the solution for this. I've been trying to find $t$ from $x$, and then ...
1
vote
2answers
81 views

How to show whether 3 planes have a common line of intersection

To show whether or not the 3 planes $$x+y-2z=5\tag 1$$ $$x-y+3z=6 \tag2$$ $$x+5y-12z=12 \tag 3$$ all have a common line of intersection. Can I do $(3)-(2)$ to get the line $6y-15z=6$ and $(1)-(2)$ ...
1
vote
2answers
18 views

Special method of solution for $A\vec x=\vec b$ where $A$ is a square matrix such that $A^tA$ is diagonal and has full rank?

Is there any special shorter method of solution other than cramer's rule for solving a system of $n$ linear equations in $n$ unknowns $A\vec x=\vec b$ where the square matrix $A$ has the property that ...
0
votes
1answer
12 views

Can we ensure convergence for the jacobi method or do we simply trial and error?

For iterative methods for solving systems of equations, we may not always get convergence and it can depend simply on the way in which we write the equations. I understand there are tests which will ...
0
votes
1answer
21 views

Need to find 3 unkown variables

In the city is $3$ types of hotels. In the $I$ type there is $150$ basic apartments and $17$ luxury apartments, in the $II$ type $310$ - basic and $37$ - luxury and in the last $III$ type $40$ - basic ...
0
votes
0answers
36 views

can it be solved without vieta formulae ??

recently i came across an anonymously remarkable algebra question which is as follows If the polynomial $$F(x)= 4x^4 - ax^3 + bx^2 - cx + 5$$ where $a,b,c$ belongs to $\mathbb R$ has 4 positive real ...
1
vote
3answers
70 views

For which values does the Matrix system have a unique solution, infinitely many solutions and no solution?

Given the system: $$\begin{align} & x+3y-3z=4 \\ & y+2z=a \\ & 2x+5y+(a^2-9)z=9 \end{align}$$ For which values of a (if any) does the system have a unique solution, infinitely many ...
0
votes
2answers
36 views

Linear Hamiltonian System

Suppose the linear system: $\dot{z} = J \frac{\partial{H}}{\partial{z}} = J S(t) z = A(t) z$, with Hamiltonian $H=H(t,z)=\frac{1}{2} z^T S(t)z$. How can I prove that: $$\frac{d}{dt}H(t,\xi(t)) = ...
1
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2answers
35 views

Finding answers to system of equations

Let's say we have such a system structure of equations: ...
1
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3answers
43 views

Systems of equations problem in algebra

A dog weighs 1/8 of a cow. Their combined total is 360kg, how much does the dog weigh and how much does the cow weigh? I got this question on my algebra test, my teacher said the answer was along ...
0
votes
2answers
49 views

Could someone explain how to solve these sets of equations please?

I am given that; $$x_1 + x_2 + x_3 = 75$$ $$x_1 + x_2 + x_4 = 75$$ $$x_1 + x_3 + x_4 = 75$$ $$x_2 + x_3 + x_4 = 75$$ I need to find $x_1, x_2, x_3$ and $x_4$. I know that each variable equals 25. ...
1
vote
0answers
34 views

How to find value of an unknown in matrix to make system of linear equations consistent

I'm currently stuck on this question relating to finding the unknown in a matrix so that the system of linear equations is consistent. I need to solve for $\lambda$. My first instinct is to try and ...
0
votes
0answers
18 views

System of linear equations with repeated equations

Suppose that I have this over-determined system of equations, $$a_1x_1 + a_2x_2 + a_3x_3 = k_1$$ $$b_1x_1 + b_2x_2 + b_3x_3 = k_2$$ $$c_1x_1 + c_2x_2 + c_3x_3 = k_3$$ $$d_1x_1 + d_2x_2 + d_3x_3 = ...
2
votes
1answer
43 views

Building a Diet Using Linear Algebra

The Question Suppose a diet calls for 7 units of fats, 9 units of protein and 16 units of carbohydrates for the main meal. Suppose the dieter has 3 possible types of food to satisfy this requirement: ...
0
votes
0answers
25 views

Numerical scheme for system of PDEs

I'm trying to solve the following coupled PDE system for my master thesis: \begin{align} \kappa_0\frac{\partial p}{\partial t}&=- \nabla \cdot v \\ \rho_0\frac{\partial v }{\partial t} &= ...
2
votes
2answers
74 views

Find numbers $a, b, c$ given that $a+b+c=12$, $a^2+b^2+c^2=50$, and $a^3+b^3+c^3=168$

Let $a+b+c=12$, $a^2+b^2+c^2=50$, and $a^3+b^3+c^3=168$. Find $a,b,c$ Suppose $a, b, c$ are roots of $P(x)$. $$P(x) = k(x - a)(x - b)(x - c)$$ But then I get $(k = 1)$ $$P(x) = x^3 - 12x^2 + ...
6
votes
6answers
82 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: $$ ...
0
votes
1answer
48 views

How to solve congruence modulo equations?

While studying Affine Cipher in cryptography it tells that we need to solve a system of modulo congruence equations. The equations are: $8\alpha+\beta\equiv 15 \pmod{26}$ $5\alpha+\beta\equiv 16 ...
0
votes
1answer
16 views

solving system of two equations

I understand up until the "this system gives" Where did he get the $u = 2v = 2(2u)=...$ line from? Also note that $k \not= 0 $here
0
votes
0answers
18 views

Multi time scales analysis on nonlinear system of ODEs

So I have this coupled set of nonlinear ODEs that I want to do a multi time scales perturbation analysis on. $ u'(t)+\frac{C \epsilon u(t)^2}{Cl}-\frac{2 \epsilon p(t)}{Cl}-\frac{2 q_1'(t)}{Cl}=0 ...
0
votes
0answers
50 views

Is it possible to resolve equations of two vectors

I have a objective function as following $$F=\int |\alpha^TG(x)-w^TJ(x)|^2 H(x)\,dx+\lambda_1 \alpha^2+\lambda_2 w^2$$ where $\alpha^T$ is transpose of vector $\alpha= \begin{bmatrix} ...
1
vote
2answers
123 views

How to solve 3 variable in 2 equation?

This paper is abstracted from 2007 British Mathematics Olympiad Round 1 Question 2. I am currently practicing grade 8 (Singapore Secondary 2) for the upcoming Singapore Mathematics Olympiad(SMO). ...
2
votes
3answers
246 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 ...
0
votes
2answers
58 views

Solving a system of equations

Solve the system of equations: $\left\{\begin{array}{l}\sqrt{2y^2-7y+10-x(y+3)}+\sqrt{y+1}=x+1\\\sqrt{y+1}+\dfrac{3}{x+1}=x+2y\end{array}\right.$ I Wolframalpha.com and get only one solution ...
1
vote
1answer
43 views

how to solve system of quadratic equations (mod N)

Given a two equations: $${(ax_1 + b)}^2 = c_1 \pmod N$$ $${(ax_2 + b)}^2 = c_2 \pmod N$$ $N=p.q$ $p$ and $q$ are large primes $x_1, x_2$ and $c_1, c_2$ are known Is it computationally feasible to ...
0
votes
3answers
79 views

Solve easy equations

Can anyone please help me with solving this equation; thanks! $$ \displaystyle \frac{d+7}{3}+4\quad=\quad -\frac{5d}{4} $$ My Steps $1.$ Multiplied both sides by $3$ and $4$ to get rid of the ...
-1
votes
1answer
57 views

Unordered pairs solution

Please help me with this question.$$$$ How many unordered triplets $(x,y,z)$ , subject to constraints, $(x^4-2x^3)_{cyclic}\leq0$ , satisfy the system of equations: ...
7
votes
1answer
190 views

Prove or Disprove the Existence of Solutions…[linear algebra] - a C.S.I.R Question

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? 1)$Ax=b$ has an integer ...
1
vote
1answer
64 views

how to work out 3 equations simultaneously

So i was doing this linear programming question and got stuck on this part, so how do you workout simultaneously $2x + 3y = 30 $ $(2/3)x + 2y = 16 $ $(16/3)x + 4y = 64$ According to lpsolve we ...
1
vote
2answers
88 views

I have the Eigenvalues, how do I get Eigenvectors?

My matrix is \begin{array}{ccc} 3 & 4 & 5 \\ -2 & 7 & 3 \\ 5 & -8 & -3 \end{array} Through the rule of Sarrus, I know (approximately) $\lambda_1 = 5.9$ $\lambda_2 = 3.5$ ...
6
votes
2answers
63 views

How to solve this nonstandard system of equations?

How to solve this system of equations $$\begin{cases} 2x^2+y^2=1,\\ x^2 + y \sqrt{1-x^2}=1+(1-y)\sqrt{x}. \end{cases}$$ I see $(0,1)$ is a root.
1
vote
0answers
31 views

Expressing the Solution to a System of Differential Equations

My professor wrote the solution to a system as $$X = C_1 \begin{bmatrix}1 \\2 \end{bmatrix} e^{\lambda_1t} + C_2 \begin{bmatrix}3 \\4 \end{bmatrix} e^{\lambda_2t}$$ Where the column vectors are the ...
0
votes
1answer
32 views

Find $a$ and $b$ in a 4 equation system

$a, b \in\mathbb{R}$. I have four equations: $$x+3y-2z+t=-3$$ $$3x+11y+az+5t=2$$ $$3x+12y-6z+6t=b$$ $$4x+15y-8z+8t=-5$$ I have to find out the values of $a$ and $b$ where the system is solvable (has ...
0
votes
3answers
44 views

Solving Linear System with inequalities

I have the following system: \begin{align} b - x = 0 \\ a - 0.33b - 0.5x =0 \\ d - 0.33b = 0 \\ a - 0.33b + c = 0 \\ a + b + c + d + 2x = 1 \\ a + b + c + d - 8.8x \le 0 \\ a + b + c + d - 7.27x ...
1
vote
0answers
33 views

Converting second order system into first order system (ODE)

The second order equation $\frac{d^2\vec{x}}{dt^2} = A\vec{x}\ + \vec{g}(t)$ models an earthquake's effect on a 7-story building. Let $x_j(t)$ be the displacement of the $j$th floor with respect to ...
1
vote
0answers
25 views

Solving a linear system in function of a parameter

Problem: Solve the following system in function of the parameter $b$: \begin{align*} \begin{cases} -bx + 2y - (2+b^2)z + bu &= -2 \\ x -2y + bz -u &= 0 \\ x + (2b-4)y + (2-b)z + (b-1)u &= ...
-1
votes
2answers
28 views

Converting a second order n x n system into a first order 2n x 2n system

Say I have the following second order 7 x 7 system of equations: $x_1'' = 10(x_2- x_1- 1)$ $x_2'' = 10(x_3- 2x_2+ x_1)$ $x_3'' = 10(x_4- 2x_3+ x_2)$ $x_4'' = 10(x_5- 2x_4+ x_3)$ $x_5'' = 10(x_6- ...