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Questions tagged [systems-of-equations]

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.

0
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5answers
35 views

Is there any solutions for this set of equation?

Is there any solutions? Why?How? $$ \begin{array}{cccc} 3z &+ &4x + 3y &= 12\\ z &+ &4x + 3y &= 12\\ 2z &+ &4x + 3y &= 12\\ \end{array} $$
1
vote
1answer
21 views

Matrixes with common parameters to result in no inverse

I've been given three matrices $A, B \ \& \ C$ which are defined as follows: $$ A = { \left[ \begin{array}{ccc} b & 5 & 8 \\ c & 1 & 3 \\ a & 4 & 3 \\ \end{array} \right]...
1
vote
1answer
14 views

How is stability for a numerical solution generalized to a system of ODEs?

Consider the system of ODEs $$y' = \begin{bmatrix}-6&4\\4&-6\end{bmatrix}y, \quad t\in[t_0, t_e], \quad y(t_0)=y_0$$ I'm asked for what stepsize the explicit Euler method generates a stable ...
-1
votes
0answers
20 views

Solving system of non linear equations. [on hold]

I have the following system of non-linear equations. $$\frac{n_{5} {\left(p_{1} + p_{2}\right)} e^{\left(l {\left(p_{1} + p_{2}\right)}\right)}}{e^{\left(l {\left(p_{1} + p_{2}\right)}\right)} - 1} -...
2
votes
1answer
30 views

Solving a trigonometric system of equations related to addition formulas.

I have the following trigonometric system of equations. $ \begin{align*} && \sin(x)\cos(y) &= \frac{1+\sqrt{3}}{4} \\ && \cos(x)\sin(y) &= \frac{-1+\sqrt{3}}{4} \\ \end{align*...
0
votes
1answer
16 views

Parametrize (as a subset of R5) the solution space of the system of equations

I am just wondering how I would parametrize the system of equations from the augmented matrix I know the following: v = -2 - 4w -3z w = 0 x = 0 y = 8 - 5z z = 0 \begin{bmatrix} 1 & 4 & -2 &...
1
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0answers
21 views

overdetermined system of non-linear equations

I have to solve an overdetermined system of non-linear equations. My system has a lot of equations, but here, for example, my system has $4$ equations. Because all the variables have to be binary, I ...
1
vote
3answers
53 views

System of quadratic equations with three variables (generic form)

Try solve a system of equation like this one. \begin{cases} (O_x -A_x)^2+(O_y-A_y)^2+(O_z-Az)^2=x^2 \\ (O_x -B_x)^2+(O_y-B_y)^2+(O_z-Bz)^2=y^2 \\ (O_x -C_x)^2+(O_y-C_y)^2+(O_z-Cz)^2=z^2 \end{cases} ...
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0answers
26 views

Solving a system of Trignometric equations

I came across this system of trigonometric equations inbetween a problem in Numerical Linear Algebra. I was required to find $p^2$ and $\cos(\theta)$ in terms of $q^{(k-1)},q^{(k)},q^{(k+1)},q^{(k+2)}$...
-1
votes
2answers
26 views

How can I prove that $10=2^{a}*3^{b}*7^{c}$ has infinite solutions?

Both in a unrescrited case and with the following restriction: $a+b+c=1$
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votes
0answers
9 views

Normal Equations (OLS) : Summation notation to system of equation form

I am struggling to "visualize" the following equation. I would like to know how the following equation can be written in system-of-equation form in order to see that there are $K$ unknowns. The ...
0
votes
0answers
31 views

System of algebraic equations concerning rates

I'm not even sure where to start on the problem. Is this possible to solve without logarithms? "A certain type of bacteria doubles every 6.5 hours. If there were 60 bacteria to start with, what is ...
0
votes
0answers
19 views

Numerical solution PDEs system

I have to solve this system of PDE: $$ \left\{ \begin{array}{c} \frac{\partial^2u_c(x,y)}{\partial x^2}+a\frac{\partial^2v_c(x,y)}{\partial x\partial y}+b\frac{\partial^2u_c(x,y)}{\partial y^2} =0\\ ...
0
votes
2answers
33 views

Characterize the values $c$ and $k$ for the system: $x+y+z=3, x + 2y +cz = 4, 2x + 3y + 2cz = k$ [on hold]

Characterize the values $c$ and $k$ for the system: $$\begin{eqnarray} cx&+y&+z&=&3\\ x& + 2y& +cz& =& 4\\ 2x& + 3y& + 2cz& =& k \end{eqnarray}$$ ...
2
votes
3answers
75 views

How to solve the system of matrix equations $XX^TA = A$, $X^TX = I$?

Given tall matrix $A \in \mathbb R^{n \times k}$ (where $n \gg k$), is there a way to solve the following system of matrix equations in $X \in \mathbb R^{n \times k}$? $$\begin{aligned} X X^T A &=...
0
votes
0answers
34 views

polynomial equation with non-integer powers

If for every $t$ $$\sum_{i=0}^{k_1}\left[\left(a_i^Tx-b_i\right)t^i\right]=0$$ where $a_i \in \mathbb{R}^{n \times 1}$, $x \in \mathbb{R}^{n \times 1}, \forall i \in \{0,\dots, k_1\}$, and $b_i \in \...
0
votes
1answer
21 views

Non-Homogeneous Differential System Yields 2 Answers

I'm trying to solve a given system: $x_1'=x_1+2x_2+5e^{4t}$ $x_2'=2x_1+x_2$ However, depending on if I apply (D-1) to the first or the second equation, then substitute in and use the annihilator ...
0
votes
0answers
45 views

How do you solve $xy=8$ and $x^y=y^x$? [duplicate]

I know answers are $4$ and $2$ but can't solve this.
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0answers
30 views

Is there an easy way to tell if a set of quadratic constraints are solvable

I am trying to find a matrix whose null space does not intersect with the column space of the matrix $$ M = \begin{bmatrix} 0 & -r_3 & r_2 \\ r_3 & 0 & -r_1 \\ -r_2 & r_1 & 0 ...
-2
votes
2answers
32 views

For which values of a the equation system has a unique solution and for which pairs (a, b) it has more than one? [closed]

Consider the following system: $$ \begin{cases} x - ay = 1 \\ ax - 4y = b \\ \end{cases} $$ For which values of a the given system has a unique solution and for which pairs of values (a, b) it has ...
0
votes
1answer
39 views

Solving system of linear equations that consists of a few matrices

I have a task to write a Matlab program that get LU-decomposition of a given matrix $A$ and then solves a system of linear equations using the obtained decomposition. $Mz = f $, where $M = \left(\...
3
votes
0answers
32 views

Solving a first order hyperbolic PDE in two variables

Question: Consider the system \begin{align} \frac{\partial u}{\partial x} + u \frac{\partial u}{\partial y} + 2\frac{\partial v}{\partial y} & = 1 \\ \frac{\partial v}{\partial x} + 2v\frac{\...
2
votes
3answers
72 views

Approximating $\sum_i \frac{a_i}{\lambda - \lambda_i} = \lambda$

I am trying to solve the equation $$ \sum_i^n \frac{a_i}{\lambda - \lambda_i} = \lambda $$ for some real constants $a_i, \lambda_i$, with $\lambda_1 > \lambda_2 > \lambda_3 > ... $ I have ...
0
votes
1answer
36 views

Find the general value of $\theta$

Find the general value of $\theta$ which satisfies the equation $(\cos\theta+i\sin\theta)(\cos3\theta+i\sin3\theta)\dots \{\cos(2n-1)\theta+i\sin(2n-1)\theta\}=1$. My attempt: $(\cos\theta+i\sin\...
0
votes
1answer
53 views

Finding multiple unknowns with significantly fewer equations [on hold]

I have 15 unknowns and only 8 equations , is there a method I can read up on that will assist me in solving this problems, I suspect matrices of some kind etc.
1
vote
0answers
31 views

Navigation via least-squares

Estimating a vector $x \in \mathbb{R}^2$ knowing its distance to four beacons $v_1, \dots, v_4\in \mathbb{R}^2$ via least-squares means finding the least-squares solution to $A x = y$, where $y\in\...
1
vote
3answers
31 views

The system of Diophantine equations with same solution

There is a system of Diophantine equations: \begin{equation*} \begin{cases} 368=x^7 (mod 407)\\ 389=x^{11}(mod 407) \end{cases} \end{equation*} However, solving each of them by hand is quite ...
0
votes
1answer
28 views

Solving simultaneous logarithmic equations from Newton's law of Cooling

A cup of warm water at $46$ degrees is placed into a refrigerator. 10 minutes later, the water is $39$ degrees, and another 10 minutes later, the water is $33$ degrees. Use Newton's law of cooling ...
1
vote
1answer
45 views

Geometrical interpretation of the solution for finding a consistent linear system

in our lecture we learned how to check if a linear system A $\cdot \vec x = \vec b$ is consistent for every $\vec b$. Example: $A=\begin{bmatrix}1&3&4\\-4&2&-6\\-3&-2&-7\...
1
vote
2answers
86 views

Finding nonnegative solutions to an equation [on hold]

Is there any method to find the nonnegative solutions of the following equation? $$ x_1^2+x_2^2+\cdots x_{10}^2=\frac{3}{4}\left(x_1+x_2+ \cdots x_{10}\right) $$ Here $x_1,x_2,\cdots, x_{10}$ are ...
0
votes
0answers
23 views

Find the set of values of b [closed]

If the quadratic formed by eliminating 𝑥 between the equations 𝑥^2+𝑎𝑥+𝑏=0 and 𝑥𝑦+𝑙(𝑥+𝑦)+𝑚=0 whose roots are the same as those of the original quadratic in 𝑥 find the set of values of b ...
0
votes
2answers
87 views

How to solve linear matrix equations?

I am new to linear algebra and I want to solve an equation. I have $A=Bx+Cy$, where $A$, $B$ and $C$ are known matrices and I want to find scalars $x$ and $y$. Is that possible through left ...
0
votes
1answer
39 views

Solving in terms of z , three variable two equation system

Solve in terms of $z$ $$ \begin{cases} 4z&= x + 2y \\ 3z^2&=\frac{1}{2}x^2 + y^2 \\ \end{cases} $$ Solution: $x = 2z/3$ and $y = 5z/3$. I don't understand how they got to the solution with ...
1
vote
1answer
102 views

Invertible solution to underdetermined system of equations

Given an underdetermined system of linear equations like $$ a X = b \tag{1} $$ with $a \in \mathbb{R}^{1 \times n}$, $X \in \mathbb{R}^{n \times n}$ and $b \in \mathbb{R}^{1 \times n}$ where $a$ and ...
1
vote
1answer
51 views

Let $I$ be a LES with its solution set $\mathcal{L}_I$ and some other solution $\mathcal{L}_2$. Is $\mathcal{L}_2\subseteq \mathcal{L}_{I}$ or not?

We have the following LES $I$ \begin{align} 3x_1-2x_2+4x_3-x_4&=1\\ -\frac43x_2-\frac{13}3x_3+\frac13x_4&=-\frac{19}3 \end{align} with the solution set $\mathcal{L}_{I}$, which contains ...
-1
votes
0answers
32 views

solve nonhomogeneous differential systems

I found this problem and I can't solve it. If anyone can help me with this , it'd be a lot appreciated. Solve the following nonhomogeneous system : $\begin{pmatrix}x\\y \end{pmatrix}^{'}=\begin{...
1
vote
3answers
93 views

Why do we need approximation methods when we have algorithms to find exact roots?

While I was studying numerical methods and optimizations recently, I observed that whenever we find a root to an equation or a system of linear equations, we always find approximate roots. However, we ...
0
votes
1answer
44 views

Least-norm solution

I am trying to find the least-norm solution of the following set of equations $$\begin{aligned} y+z &= -3\\ x+2y+z &= -2\\ -2x-3y-z &=1\end{aligned}$$ Using the expression for the ...
0
votes
1answer
39 views

Finding the value of $ k $.

Find all real value of $ k $ such that the system of equation \begin{align} a^2 + ab &= kc^2 \\ b^2 + bc &= ka^2 \\ c^2 + ca &= kb^2 \end{align} have positive real number solution for $ a $...
0
votes
1answer
71 views

Solving a system of 3 linear inequalities with 3 unknowns

find lowest possible x, y and z whole number variables where: x =< 2y+2z 6y =< x+z 3z =< x+y I am trying to solve this system of 3 linear equations ...
1
vote
0answers
13 views

Provide a geometrical interpretation of span vector

My question is about: Find the conditions onto $b_1, b_2$, and $b_3$ such that the following system is consistent: $2x_1 - 2x_2 = b_1$ $3x_1 + 3x_2 = b_2$ $4x_1 - 4x_2 = b_3$ Provide a geometrical ...
0
votes
5answers
63 views

What is $\log_{a}{x} \cdot \log_{y}{a}$ given below system of equations?

I let $\log_{a}{x}=m$ and $\log_{y}{a}=n$. So I have to find $m\cdot n$. From the system of equations we get $$m-\frac{1}{n}=1 \quad \quad n-\frac{1}{m}=1$$ From here I find that $m=n$ (...
2
votes
0answers
21 views

Reference on higher-order coupled system of ODEs

I have the system of $m$ coupled ODEs given as: $$A_4 x^{(4)} + A_3 x^{(3)} + A_2 \ddot{x} + A_1\dot{x}+A_0x =0$$ along with $4m$ initial (i) and final (f) conditions $x_i$, $\dot{x}_i$, $x_f$, ...
0
votes
1answer
26 views

Solve Ax=b using Cholesky decomposition

I was reading the following article about the direct stiffness method. When it comes to solving the system of equations: The site states: [...]There are several different methods available for ...
5
votes
5answers
305 views

Quickest way to find $a^5+b^5+c^5$ given that $a+b+c=1$, $a^2+b^2+c^2=2$ and $a^3+b^3+c^3=3$

$$\text{If}\ \cases{a+b+c=1 \\ a^2+b^2+c^2=2 \\a^3+b^3+c^3=3} \text{then}\ a^5+b^5+c^5= \ ?$$ A YouTuber solved this problem recently and, though he spent some time explaining it, took over 40 ...
0
votes
1answer
28 views

Explanation: System of quadratic equations, solved incorrectly through algebra

I'm quite young and haven't used this site before, so I apologise if I'm not laying things out correctly or any other problem really (tips would be appreciated). Regardless, here's something I'm quite ...
0
votes
1answer
35 views

Eigenvalues of a block matrix from the eigenvalues of the blocks

I am trying to find the eigenvalues of the following complex matrix \begin{align} M=\left(\begin{matrix} A & B \\ B^\dagger & A^\dagger \end{matrix}\right) \end{align} where the symbol $\...
1
vote
3answers
42 views

Linear system with constants in arithmetic progression

I have stumbled upon a mathematical problem, that could be translated into English like this: There's a system of n linear equations with n unknowns, and with constants that are in arithmetic ...
1
vote
1answer
20 views

If the row echelon form of a linear equation system has a line with zero entries, then it has more than a solution.

If the row echelon form of a linear equation system has a line with zero entries, then it has more than one solution. Let \begin{align} \text{I}& \qquad 3x_1&=2\\ \text{II}& \qquad 0x_1&...
1
vote
4answers
54 views

x^2+y^2=2, xy=1, how to find x and y

I have problems when doing these equations when I don't know any variable's value. Can someone please explain how to do this and possibly give some tips when it comes to solving these problems? Well, ...