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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6
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4answers
58 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
1answer
58 views

What do we know about the solution of this set of linear equations?

Let $C \in \mathbb{R}^n$, $A \in \mathbb{R}^{n \times n}$, $Y \in \mathcal{Y}$, and $B : \mathcal{Y} \to \mathbb{R}^n$ be linear, where the linear space $\mathcal{Y} \subset \mathbb{R}^m$ may be ...
1
vote
2answers
27 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
31 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 ...
1
vote
1answer
38 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
3answers
69 views

Matrix Problem of form Ax=B

The matrix $A$ is given by $$\left(\begin{array}{ccc} 1 & 2 & 3 & 4\\ 3 & 8 & 11 & 8\\ 1 & 3 & 4 & \lambda\\ \lambda & 5 & 7 & 6\end{array} \right)$$ ...
2
votes
3answers
228 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 ...
2
votes
1answer
17 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
14 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
22 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
1answer
299 views

solving a non-linear (trigonometric) system of equations with two equations and two variables

I'm trying to solve the following system of equations: $$l_1*sin(\alpha)=l_2*cos(\gamma)+l_3*sin(\beta)$$ $$l_2*sin(\gamma)+l_1*cos(\alpha)=l_3*cos(\beta)+l_4$$ with the unknowns $\beta$, $\gamma$ ...
1
vote
3answers
43 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
47 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
17 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
18 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
53 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 ...
3
votes
2answers
74 views

For what values of k and h does this system of equations have a unique solution?

Here's my system of equations: $x−3y+2z=5$ $2x−5y−3z=9$ $−x−y+kz=h$ So I have $ \begin{bmatrix} 1 & -3 & 2 & 5 \\\\ 2 & -5 & -3 & 9 \\\\ -1 & -1 & k& h ...
3
votes
2answers
796 views

Determine the values of $k$ so that the following linear system has unique, infinite and no solutions.

Determine the values of $k$ so that the following linear system has a unique solution, infinite solutions and no solution. $2x + (k + 1)y + 2z = 3$ $2x + 3y + kz = 3$ $3x + 3y − 3z = 3$ I have ...
0
votes
0answers
13 views

Find the eigenvalues/vectors for the $N\times N$ matrix C summarizing the system $-X_{t-1} +2X_{t} - X_{t+1} = \lambda X_{t}$ [on hold]

Consider the system $-X_{t-1} +2X_{t} - X_{t+1} = \lambda X_{t}$ for $k = 1,...,N$ Write down the NxN matrix C such that the above equations are equivalent to $CX = \lambda X$. Find the ...
0
votes
1answer
27 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)) = ...
6
votes
1answer
164 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
2answers
33 views

Finding answers to system of equations

Let's say we have such a system structure of equations: ...
1
vote
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
1answer
39 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
1answer
39 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 ...
2
votes
0answers
52 views

Solving the algebraic equations .

I am working with an equation to find the singular points in $\mathbb P^2 (\mathbb C)$ . Basically after taking the partial derivatives and doing some manipulations it reduces to $$y^2 + (2-k)xz ...
1
vote
1answer
107 views

Cramer's Rule with complex system of equations

Given a 2x2 system of complex equations with one unknown, $z$, written as a 2x2 matrix, $A$, would the system have infinitely many solutions iff $\det(A_x)=\det(A_y)=\det(A)=0$? Or is there more to ...
1
vote
0answers
19 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
2answers
94 views

Solutions for $-x=\textrm{tan}(x)$

How can the following equation be solved? $-x=\frac{\sin(x)}{\cos(x)}$, $x \in \mathbb{R}$ I understand $x=0$ is one solution, but I need all of them (In particular, precisely those that are not ...
-1
votes
1answer
22 views

When do 2 cars even out [closed]

I have two different cars. $A$ cost \$17,325. $B$ cost \$21,640. $A$ gets $34$ mpg. $B$ gets $46$ mpg. But, Fuel for $A$ cost \$2.40 a gallon & fuel for $B$ cost \$3.60 a gallon. If these costs ...
0
votes
0answers
17 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 = ...
6
votes
6answers
74 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: $$ ...
2
votes
1answer
32 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
21 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
55 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 + ...
7
votes
4answers
174 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
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} ...
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 ...
4
votes
1answer
98 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
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: ...
1
vote
2answers
58 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). ...
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 ...
9
votes
2answers
119 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 ...
0
votes
3answers
69 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 ...
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.
0
votes
2answers
55 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, ...