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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.

2
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0answers
19 views

Attempting to simplify a quadruple system

I am attempting to simplify a horribly complicated equation that can be expressed in the form $\frac{ay^2}{(b(x-c))^d)}=z$. By plugging in x and y values I am able to find points to set up a system ...
2
votes
1answer
58 views

Solving efficiently $Ax=b$

Calculating efficiently $$AX = C$$ Given $$ A = I + BB^{T} $$ where $ B $ is an orthogonal matrix of $ n \times m $. Since $B$ is a semi-orthogonal matrix, does $ BB^{T} = I_n $ hold? If not, how ...
1
vote
1answer
37 views

Equations of Motion in Cylindrical Co-ordinates

I've run into an interesting set of differential equations, that I'm not 100% sure where to begin- I'm not looking for a 100% complete solution, more just a push in the right direction of where I can ...
1
vote
2answers
28 views

Linear system of equations that arises from recurrence relation.

So when solving a third order homogenous recurrence relation I end upp with the task of determening the constants $a,b$ and $c$. The initial conditions give rise to the following system of equations: ...
0
votes
2answers
31 views

How to solve for two variables $10x + 20y \geq 10203$

I have a few equations of the form $$10x + 20y \geq 10203$$ Basically just two variables set equal to a value. Each equation is unrelated. So basically the goal is to find an x and y that is equal ...
0
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0answers
17 views

Show validity of Dulac criterion in not simply connected region

can anyone suggest me a not simply connected region where Dulac criterion is still valid? When I think on a not simply connected region the only thing that comes to my mind is an annulus, but in an ...
0
votes
2answers
17 views

What is the rule to “guess” how to multiply both equations of a system so that their sum solves in “good” (perfect squares) numbers?

How did the author guess from the beginning that 1st equation must be multiplied by $3$ and the 2nd - by $17$?
0
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1answer
22 views

Where is a mistake equating $x$ derived from both equations of a system?

I know how to solve this, but why is the below reasoning wrong and leads to a mistake (I don't see any mistake!) Step 1: From first equation $x=\dfrac{8}{y}$ , and $y$ is not zero Step 2: From ...
0
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2answers
43 views

how to solve the equation $\dfrac{a}{x-a}+\dfrac{a}{y-a}$ is [on hold]

Please provide the steps to solve the equation, the answer to this equation is zero I am not sure how it is derived, Kindly help if $x+y=29$ then the value of $\dfrac{a}{x-a}+\dfrac{a}{y-a}$ is
0
votes
1answer
36 views

Simultaneous equations involving exponentials

From the following simultaneous equations: \begin{align} e^{(m+\frac{1}{2}s^2)}=log S^{(1)}(0)+log S^{(2)}(0)+2rT-\bar{V_1}\frac{T}{2}-\bar{V_2}\frac{T}{2}\\ e^{(2m+s^2)}(e^{s^2}-1)=\bar{V_1}T+\bar{...
3
votes
3answers
171 views

Exponential/Logarithmic equation system

Solve the following equation system over the real numbers $$\begin{cases} x(1-\log_{10}(5))=\log_{10}(11-3^y)\\ \log_{10}(35-4^x)=y\log_{10}(9) \\ \end{cases} $$ For the functions in the above ...
0
votes
2answers
54 views

How to solve this differential equation 2

\begin{align} \ x\frac{dy}{dx}-4y &= x^6e^x \\\\ \end{align} By dividing on x we get : \begin{align} \ \frac{dy}{dx}-4\frac{y}{x}=x^5e^x \end{align} Now I got the integrating factor"by e^...
0
votes
0answers
30 views

Infinite solutions of system always between 0 and 1

i would like know if there is an algorithm that solve a system with infinite solutions z = Wa where W is the matrix of coefficents and i want find z, i know there are a lot of algorithms that permit ...
1
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2answers
38 views

Solving a system of equations with an unknown constant

How do I find all values of: $$ a \in R $$ for the system of equations below? $$ \left\{ \begin{array}{c} x_2-3x_3+2x_4=1 \\ x_1+3x_2-x_3-x_4=9 \\ x_1+4x_2-4x_3+(7+a)x_4=16+a \\ x_1-2x_2+14x_3-...
0
votes
3answers
32 views

The real solution from a system of equation

I found this question from my friend's math competition, I don't know where I must start it There are 3 couples of real numbers $$(x_1,y_1) (x_2,y_2)$$ and $$(x_3, y_3)$$ that satisfies the system of ...
2
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0answers
21 views

Is there a technique to simplify a system of polynomial equations using known solutions?

If I am working with a polynomial and guess a root $x_0$, I can divide the polynomial by $x-x_0$ to obtain a simpler polynomial without that one root. Can I do something similar for a system of ...
-1
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2answers
28 views

Solve equation to find two missing numbers in a sequence of positive integers

This problem is inspired from an algorithm question at https://stackoverflow.com/questions/3492302/easy-interview-question-got-harder-given-numbers-1-100-find-the-missing-numbe?rq=1. Summary of ...
1
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1answer
48 views

Solving equation of degree 5

Find all real value of $x$ such that $$x^2+1=2\sqrt[3]{2x-1}$$ Let $t=\sqrt[3]{2x-1}$. Then the equation is equivalent to $$\left(\frac{t^3+1}{2}\right)^2+1-2t=0 \Leftrightarrow t^6+2t^3-8t+5=0 \...
1
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0answers
34 views

Is there something substantial behind this solution technique?

In this video, the two variable equation $(1 - \alpha - \frac{\gamma}{2})\vec A + (\frac{\alpha}{2} - \frac{\gamma}{2})\vec B + (-1 + \frac{\alpha}{2} + \gamma)\vec C = \vec 0$ is solved by ...
0
votes
1answer
15 views

System of equations. Find values for A.

Find the values of the parameter a for which the system has (1) one solution, (2) no solutions, and (3) infinitely many solutions. In case (3), find the solution. I'm so sorry about the picture, but ...
2
votes
1answer
23 views

How to transform this third order nonlinear differential equation in to a first order system of differential equations

Transform the differential equation $$ \begin{cases} u'''(t) = \sin(u''(t)) - u^2(t), & t > 0, \\ u^{(i)} = u_i \quad \text{for } i \in \{0, 1, 2\} \end{cases} $$ into an equivalent system ...
0
votes
1answer
40 views

How to find solution of these equations?

$$\begin{cases}x_1+y_1=a\\ x_1-y_2=b\\ y_1-x_2=c\\ x_2+y_2=d \end{cases}$$ How to find a solution for these equations because from elimination and substitution method I end up with 2 equations with ...
0
votes
1answer
11 views

Solving equation where the independent terms' vector is defined in function of the variable vector.

I'm trying to teach myself linear algebra for and one of the exercises of the book I'm reading is the following: Find all solutions in $\vec x= \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} ∈\...
1
vote
0answers
20 views

How to find the number of machines given a set of rental prices?

I'm stuck in a situation involving rental. Typically I would assign a set of constants based on their price but I'm confused on how to use it or if it applies here. The problem is as follows: A tv ...
-1
votes
1answer
45 views

Writing Short Equations/Equivalents For A Group Of Numbers.

I have a series of numbers between 0 to 16,000,000. It's certainly possible to describe some (if not all) of these numbers with some equations. For example, it's possible to write 720 as 6! and 5040 ...
0
votes
3answers
30 views

Find the values of $a$, for which this system of linear equations has one solution, no solution, or infinite solutions

Determine the values of a for which the following system of linear equations has no solutions, a unique solution, or infinitely many solutions. You can select 'always', 'never', '$a = $', or '$a \neq$'...
0
votes
1answer
22 views

Getting the best estimative for $Ax = B$ with A being a non squared matrix

Problem: I'm analyzing an enterprise for the purpose of investment and I need to find out the average cost of opening a single store. The struggle here is that the company has 3 different brands, so ...
1
vote
1answer
40 views

Some interesting systems of equations [closed]

1, Solve the system of equations:$\left\{\begin{matrix} x^3+y^3+2z^3=19x-11y-5z+1\\ x^3+(y^2+1)x=(x^2+y^2)z+z \\ \sqrt{2+x^2+y^2-2yz}=y^2+z^2-2xy+\sqrt{2} \end{matrix}\right.$ 2,Solve the system of ...
0
votes
0answers
19 views

Iterative method for a system of linear equations

I need to solve this problem: Let $A\in \mathcal M_n(\mathbb R)$ a positive definite matrix, $B \in \mathbb R^n $, and $AX=B$ a system of linear equations. We use an iterative method which ...
1
vote
0answers
28 views

Two Equations and Two Unknowns

$$\frac { - \left( 2 e ^ { h + l } \left( 2 ( - c + l - 1 ) e ^ { h + l + l } + ( n - 2 ) ( - c + l - 1 ) e ^ { h + l } + n ( - c + l - 1 ) e ^ { l + q + l } - 2 e ^ { h + l } \right) \right) } { \...
0
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0answers
19 views

Why is the Bellman principle of optimality considered a system of linear equations?

I am currently trying to understand why the Bellman Principle of Optimality is considered a system of linear equations. The Bellman optimality equation, taken from Reinforcement Learning - An ...
0
votes
3answers
60 views

How to get only positive solution of a system of 4 variables and equations?

I have a system of $4$ equations in $4$ variables: \begin{align} x_1 + y_1 &= m\\ x_2 - y_1 &= n\\ x_1 - y_2& = o\\ x_2 + y_2 &= p\end{align} $x_1, y_1, x_2, y_2$ are ...
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0answers
24 views

If the production cost decreases by 20% and the selling price remains the same, calculate the profit in usd.

I am stuck on this math. Please help me to get rid of this problem. When the per unit production cost of an item is usd. 80 and other costs per unit is usd. 20, the producer makes a profit of 15% on ...
1
vote
1answer
19 views

Equilibrium point of a linear system of DE's

I have the following linear system: $$\dot{x} = \sigma - kx$$ $$\dot{y} = -kx+k_1(y_m-y)$$ With $k,k_1,\sigma,y_m\in \mathbb{R}^+$. (That is, all strictly positive constants.) My textbook claims ...
-1
votes
1answer
49 views

Explicit expression for solutions $(x,y)$ of Diophantine equation $ax+by=d$.

Given $a,b\in\mathbb{Z}$. It is known that $\gcd (a,b)=d$ implies $$\exists x,y\in\mathbb{Z}, \ ax + by=d .$$ I have been looking for an explicit expression of any solution $(x,y)$, in terms of $a,b,...
0
votes
1answer
44 views

Algebra precalculus problem

I need to solve this problem and I don’t know how. If $y^2 + z^2 + yz = a^2$ $z^2 + x^2 + zx = b^2$ $x^2 + y^2 + xy = c^2$ $yz + zx + xy = 0,$ then $a \pm b \pm c = 0$ I can see that $a^2 + b^2 +...
1
vote
2answers
39 views

Problem solving a 4 by 4 linear system with infinite solutions

I have a question here. The following is a 4 by 4 system of linear equations which has infinitely many solutions. $$ \left\{ \begin{array}{c} x+y+3z+t=0 \\ x-y-z-t=0 \\ 3x+y+5z+3t=0 \\ x+5y+11z+...
-7
votes
1answer
43 views

Solve for numbers that satisfies the equation [closed]

if $x^2+y^2=29$ and $x^3-y^3=121$ What are the values of $x$ and $y$ ?
2
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0answers
34 views

How to solve this set of integral equations? [closed]

Given some known functions $f(x)$ and $g(x)$, is it possible to find all solutions $\psi(x)$ that satisfy the following constraints? $$1=\int_{-\infty}^\infty dx\,f(x)\psi(x)$$ $$1=\int_{-\infty}^\...
0
votes
0answers
17 views

For which value(s) of $p$ and $q$ does the system have a) infinitely many solutions and b) a unique solution?

The system is as follows: $x+y+z=5$ $2x+py+qz=4$ I would be eternally grateful if someone could help me with this question!
0
votes
0answers
10 views

Find the minimum and maximum values for variables from two equations

I'm looking at a problem where the following two equations are given: $$ xy \ge C $$ and $$ a \le {\frac{x}{y}} \le b, $$ where $x> 0, y >0$. Then they say at which values min/max values for $...
1
vote
1answer
60 views

Banach fixed-point theorem : Existence of solution

We have the system \begin{align*}&x_1=\left (5+x_1^2+x_2^2\right )^{-1} \\ &x_2=\left (x_1+x_2\right )^{\frac{1}{4}}\end{align*} and the set $G=\{\vec{x}\in \mathbb{R}^2: \|\vec{x}-\vec{c}\|_{\...
0
votes
1answer
27 views

Can you help me eliminate the variables?

Hi can you help me solve this exercise? Thanks. Eliminate $x, y, z$ from the given equations $$ \left\{ \begin{array}{ll} bx^2 +lx + c &= 0\\ cy^2 + my + a &= 0\\ az^2 + nz + b &= 0\\ xyz ...
0
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1answer
39 views

Need help to determine basic solutions of the system of linear equations

I'm currently stuck in determining the basic solutions for the following system of linear equations, would anyone be kind enough to determine it for me? Thanks!
1
vote
1answer
44 views

Precalculus algebra problem

I need to solve this problem and I don’t know how: if $x + \frac 1y = y + \frac 1z = z + \frac 1x$, then $x=y=z$ or $x^2y^2z^2 = 1.$ I don’t need a full solution -- just a hint -- because I really ...
2
votes
1answer
50 views

When is a matrix similar to $\begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix}$

Suppose we have a matrix $A \in M_2(\mathbb{C})$ such that it's characteristic polynomial is $p_{A}(t) = t^2$. Prove that $A$ is either similar to the zero matrix or similar to $\begin{bmatrix} 0 &...
0
votes
1answer
50 views

A simple euclidean geometry problem of angles of a triangle using linear equations [duplicate]

I am having problems solving this problem: Using only basic geometry is easy to go here: And propose 4 equations: $$ x+y+70=180$$ $$x+w+40=180 $$ $$u+y+50=180 $$ $$u+w+20=180 $$ And it doesn't make ...
0
votes
1answer
21 views

Solving System on Inequalities Numerically

I am stuck on a problem that says given for the range [750,1800] (where each element has a spacing of 5 units: [750,755,760,...,1795,1800]), to select 4 elements from this range who are all at least ...
4
votes
1answer
33 views

The recursive systems of equations with the following form, have they ever been considered?

Suppose a system of 2 equations defined as: $$\begin{cases} x_{n+1}&=f_x(x_n,y_n) \\ y_{n+1}&=f_y(x_n,y_n) \end{cases}$$ where initial conditions for $x_0$ and $y_0$ are defined, and $x_n,...
0
votes
2answers
50 views

What is the value of the variables in the equation?

$$x_1 + y_1 = 3$$ $$y_1 - x_3 = -1$$ $$x_3 + y_3 = 7$$ $$x_1 - y_3 = -3$$ Find the values of $x_1$, $x_3$, $y_1$ and $y_3$. All I am getting is $x_1 + x_3 = 4$ and $y_1 + y_3 = 6$.