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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2
votes
2answers
57 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
43 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
14 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
16 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
47 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
64 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
236 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
54 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
41 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
71 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
54 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
175 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
61 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
85 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) $\lambda_1 = 5.9$ $\lambda_2 = 3.5$ ...
-3
votes
0answers
29 views

need to solve $AX=B$ analytically

I have to solve the following system of equations \begin{equation*} a1x+b1y+c1z+d1t=e1\\ a2x+b2y+c2z+d2t=e2\\ a3x+b3y+c3z+d3t=e3\\ a4x+b4x+c4z+d4t=e4 \end{equation*} where ...
6
votes
2answers
56 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
28 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
31 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
43 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 ...
0
votes
2answers
56 views

Rearranging a system of trigonometric equations [closed]

I have the following two equations: $$\begin{align}17\,t\cos\theta &= x + 8\,t\sin\alpha \\ 17\,t\sin\theta &= y + 8\,t\sin\alpha.\end{align}$$ where $x$, $y$ and $\alpha$ are known values, ...
1
vote
0answers
30 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
23 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
25 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- ...
0
votes
0answers
38 views

Solving a general system of linear equations

We are given a system with n linear equation: $$\forall i\in \{1,...,n\}: i \cdot x_i + \sum_{j=i+1}^{n}x_j= \frac{i}{n}$$ Prove that the solution for this system of equation is $$\forall i\in ...
4
votes
1answer
101 views

How to determine when this two variable transformation is invertible?

I am given: $$ X= U \cos(V) \tag{1}\\ $$ $$ Y = U \sin V \tag{2}$$ Now, I need to: a) Give the respective ranges for $U$ and $V$ in order that the transformation defined is one to one. and ...
-1
votes
1answer
41 views

Algebra - Solutions of linear systems

How would I find the real values of $k$ such that the following linear system does not have a unique solution? $$\begin{cases} x + 3y + kz = a \\ 2x + (2k+2)y + (3k-2)z = b \\ kx + (k+4)y + 4z = c ...
0
votes
3answers
87 views

Solve this equation: $7x^3+11x^2+6x+2=(x+2)\sqrt[3]{x^3+3x^2+5x+1}$

Solve this equation: $7x^3+11x^2+6x+2=(x+2)\sqrt[3]{x^3+3x^2+5x+1}$ I used Wolframalpha.com and get solutions $x\in\left\{0; \dfrac{-9\pm\sqrt{53}}{14}\right\}$. But I can't solve this.
1
vote
1answer
35 views

The Lotka-Volterra Model Continued

Consider the following system of equations, and assume that population of prey is measured in thousands, and that the population of predators is measured in hundreds. $$\frac{dx}{dt} = ...
1
vote
1answer
72 views

Predator Prey Model

Consider the following system of equations, and assume that population of prey is measured in thousands, and that the population of predators is measured in hundreds. $$\frac{dx}{dt} = ...
0
votes
1answer
37 views

Solve system of trigonometric equations

How would you solve a system like this $$ \left\{ \begin{aligned} 0&=E-\sin\left(\theta_1\right) + K \sin\left(\theta_2 - \theta_1\right)\\ 0&=E+\sin\left(\theta_2\right) - K ...
1
vote
4answers
60 views

Why would I divide these two equations to solve for i?

I have the following two equations representing a longer actuarial practice question. I properly set up the equations, but am stumped on how to solve them. The book says to divide the first by the ...
3
votes
2answers
41 views

How to solve the equations of linear combination of sigmoid functions?

Let $\sigma(x)=\frac{1}{1+e^{-x}}$ be the sigmoid function. How to solve such kind of equations? \begin{align*} \sigma(x+y)+\sigma(x-y)=a\\ \sigma(2x+y)+3\sigma(3x-y)=b\\ \end{align*} I guess this ...
0
votes
1answer
48 views

Non computational approach to this equation?

I was thinking about the following problem (not homework): Let $a,b,c,d \in {0,1,2,3,4,5,6,7,8,9}$ Find all four digit numbers $abcd$ where the two digit numbers $$ ...
0
votes
1answer
24 views

Solving the equations .

Say , I have two equations : $$y_1=a+bx_{1}+e_1$$ $$y_2=a+bx_{2}+e_2$$ Say , $a=.5$ , $b=2.1$ , $x_1=2$ , $x_2=2.2$ . Now if $e_1=e_2$ , I have to find the relationship between $y_1$ and $y_2$ . ...
0
votes
1answer
70 views

Are there any system(s) of mathematics whose relationship between variables bears difference to that found within mainstream mathematics?

I have been reading up on boolean algebra quite recently, for those not familiar, this type of mathematical system has much to do with the way logic is represented (and is primarily applied to, though ...
0
votes
1answer
27 views

Solving a system of equations containing complex numbers - Gaussian elimination

Problem: Determine the solutions in $\mathbb{C}^3$ of the following system over $\mathbb{C}$: \begin{align*} \begin{cases} 2x+iy-(1+i)z &=1 \\ x-2y+ iz &= 0 \\ -ix +y -(2-i)z &= 1 ...
0
votes
1answer
10 views

Different results while calculating eigenvectors with Gaussian elemination

Regarding this matrix $\begin{matrix} 1 & 1 \\ 1 &-1 \\ \end{matrix}$. In the end I have to solve this equation system: $(\sqrt2-1)x_1-x_2=0$ $-x_1+(\sqrt2+1)x_2=0$ While the ...
1
vote
0answers
52 views

Does this linear system of 5 unknowns and 2 equations have multiple solutions?

\begin{cases} x+ 2y - z + w - t = 0 \\ x - y + z + 3w - 2t = 0 \end{cases} Add 1st to the 2nd: $$2x + y + 4w - 3t = 0 \\ y = -2x - 4w + 3t = 0$$ Substitute y in the 1st: $$x - 4x - 8w + 6t - z ...
1
vote
1answer
36 views

How to describe behavior of population system, given by system of ODEs?

So I have a system of equations:$$x'(t)=x(t)(4-2x(t)-y(t))\\y'(t)=y(t)(3-x(t)-y(t)) $$ What I understand so far is: if we have x being the population of prey and y is the population of predators. x ...
9
votes
2answers
120 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 ...
1
vote
1answer
23 views

Why do the 1's in Gauss Jordan RREF need to be along main diagonal and not other diagonal?

I've practiced G-J elimination and understand most of the algorithm insofar as it represents the different manipulations one can apply to a system of equations. However, when we're talking about ...
-2
votes
1answer
41 views

Trouble with two equations with 4 unknowns [closed]

I was wondering if I could receive assistance for the following system: $$\begin{cases}(x/a)^{3.2}+(y/b)^{3.2}=1\\ a/b = 174.1/86\end{cases}$$ I'm looking for integer solutions or how to find them ...
2
votes
0answers
17 views

Function intersecting 3 points & deriviate is positive for a range of x values

Thank you for taking the time to help out on this question. I'm looking for a function that intersects 3 points, and a derivative for every value of x between x=0 and x = 365 where dy/dx >= 0. My ...
0
votes
0answers
17 views

System of equations and chain rule

I have this system $\nabla b(z_1,\ldots,z_m)=\psi^\prime(\theta)\nabla\theta(z_1,\ldots,z_m)$. Where we have $\theta(z_1,\ldots,z_m)=\sum_i^m z_i$ and $b(z_1,\ldots,z_m)$=$\sum_i^m z_i^2$ What is ...
0
votes
1answer
18 views

Discrete convolution equation

Let $x_1 = (x_1^k)_{k =-\infty}^{+\infty}$, $x_2 = (x_2^k)_{k=-\infty}^{+\infty}$, $x_3 = (x_3^k)_{k=-\infty}^{+\infty}$ be three sequences of real numbers such that $x_j^k = 0$ for $k < -m_j < ...
1
vote
1answer
46 views

Is it possible to solve this system of equations? [duplicate]

Consider a system of equations given below: $ p_1 + p_2 + p_3 + p_4 + p_5 = 1 $ $ x_1*p_1 + x_2*p_2 + x_3*p_3 +x_4*p_4 =0$ $ x_1^2*p_1 + x_2^2*p_2 + x_3^2*p_3 +x_4^2*p_4 =1$ $ x_1^3*p_1 + ...
0
votes
1answer
18 views

tridiagonal matrix with a corner entry from upper diagonal

I am trying a construct a matlab code such that it will solve an almost tridiagonal matrix. The input I want to put in is the main diagonal (a), the upper diagonal (b) and the lower diagonal and the ...
0
votes
2answers
26 views

Is it possible to have a system of equations that all equal 0, and not have each unknown's value be 0?

I'm doing about a 2 hour long homework assignment where by hand I must construct a 10x10 matrix representing a system of equations. Based on the pattern I'm seeing, I can tell all of the equations ...
10
votes
1answer
295 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 $(a,b,c)$ ...