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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.

1
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3answers
34 views

Determine if a system of equations is independent, dependent or inconsistent

Is there a way to determine the nature of a system of equations without solving it? For example, given the system \begin{cases} 2x + y - 4z = 6 \\[4px] y - 2z = 2 \\[4px] 4x + 3y - 10z = -...
0
votes
1answer
24 views

Proving a system of equations has solutions based on knowing the determinant

Once again I'm facing a problem I'm not sure I can solve. I did some messing around with the system of equations and got the formula that I wanted, but I'm unsure about the assumptions I made. If you ...
0
votes
1answer
29 views

Solve the following system of linear congruences : $x\equiv 4\pmod{12}, \quad x\equiv 7\pmod{21}, \quad x\equiv 10\pmod{15}$ [on hold]

Question: Solve the following system of linear congruences: $x\equiv 4\pmod{12}, \quad x\equiv 7\pmod{21}, \quad x\equiv 10\pmod{15}$ Here 12, 21, 15 are not pairwise coprime. How can I apply the ...
1
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0answers
21 views

Linear optimization of polynomials

Hi I want to solve the following exercise: Given $v_1,...,v_6$ in $K^2$, and x = $\sum_{l=1}^6 \frac{l}{21} v_l$, show that there exist $\{i,j,k\} \in \{1,...,6\}$ and $\lambda_i,\lambda_j,\...
0
votes
3answers
31 views

How to find the solution given below through this system?

Knowing that $a^2 + b^2 = c^2 + d^2 = e^2 + f^2$ the following equalities are given $$ac + bd = ec + df = ae + bf$$ A solution to this equality is given if $a=-c-e$ and $b=-d-f$. From these ...
0
votes
2answers
48 views

Basin of Attraction of simple nonlinear coupled ODE

Consider ($\epsilon = 0.1$) \begin{equation}\label{eq:general eq} \begin{aligned} \dot{x}_1(t) &= x_1(x_1-0.5)(x_1+0.5)+\epsilon x_2\\ \dot{x}_2(t) &= x_2(x_2-0.5)(x_2+0.5)+\epsilon x_1 \end{...
0
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0answers
14 views

Inversely proportional score based on profit margins

My question might be pretty simple but I'm having a hard time to solve it. I'm trying to create a score based on companies profit margins. The lower the profit margins, the higher the score. That ...
2
votes
1answer
17 views

Systems of Absolute Inequalities Answer Format

I was doing some online questions and ran into this problem: (the following is a system) $|3-2x|\ge1$ $x^2/(x+2)\le0$ The computer wouldn't accept the answer $x<-2$ nor would it accept the ...
4
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2answers
97 views

How can I solve $x^\sqrt{y} +y^\sqrt{x} =\dfrac{49}{48}$ and $\sqrt{x}+\sqrt{y} =\dfrac72$?

I have tried Wolfram Alpha and Mathematica to get the solution of the below system, but no result , I have used variable change $z=\sqrt{x}+\sqrt{y}$ for simplification but no result , $$ \left\{ \...
3
votes
1answer
26 views

How to properly choose a solution to a system of equations of trigonometric functions?

I worked my way to encounter a system of equations \begin{equation} \begin{cases} q=k_1\cos\phi_1+k_2\cos\phi_2& \\ 0=k_1\sin\phi_1+k_2\sin\phi_2& \end{cases} \end{equation} ...
1
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1answer
42 views

Solution of a equation of matrices

Can I find an explicit solution for the $R$ satisfying the equation $$\sum \limits_{k=1}^{\infty}R^kS(R^k)^T = M,$$ where $S$ and $M$ are known real, symmetric, and square matrices. Any help will be ...
1
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1answer
70 views

Modular solution to $ax - by \equiv 0\pmod{p}$?

Given prime $p$, integers $x$ and $y$ where both $x, y < p$ and $x \neq y$, is there an efficient way to find nontrivial coefficients $a, b$ where $a, b < \sqrt p$ such that $$ax - by \equiv 0\...
0
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1answer
29 views

Finding a solution to a linear system with more unknowns than equations subject (in R)

I have a large system of equations in the software R, with more unknowns than equations. This obviously have a (unlimited) range of solutions. An example of such a system would be $x_1 + x_2 + x_3 = ...
0
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0answers
63 views

Problem of Arithmetic Equation

I was trying to prove the Beal's Conjecture by using Arithmetic Progression but suddenly I got a new idea but after getting the result of that idea I can't able to get the solution of the equation I ...
0
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1answer
29 views

Is there any condition for a 2x2 system of equations to have a positive solution?

A $2$x$2$ system of heterogeneous equations has a unique solution if and only if the determinant of the matrix of its coefficients be non-zero. There is a $2$x$2$ system of heterogeneous equations ...
2
votes
1answer
44 views

Card shuffling and harmonic numbers

Below is a simple proof of one connection between card shuffling and harmonic numbers. I'm interested in references for this if it's already known, as well as alternative methods of proof. (Can the ...
3
votes
1answer
94 views
+50

Find the geometry of the curves of the contour lines of $f(x) = \frac{1}{2}x^tAx + b^tx + c$

Find the geometry of the curves of the contour lines of a quadratic function $$f(x) = \frac{1}{2}x^tAx + b^tx + c$$ where $A \in \mathbb{R}^{2 \times 2}$, $b\in \mathbb{R}^2$ and $c\in \mathbb{R}$ ...
0
votes
0answers
40 views

Solve the following equation to find value of $64xyz$ [on hold]

If the real numbers $x,y,z$ are such that $x^2 + 4y^2 + 16z^2 = 6xy + 4yz + 2zx =3$. Then find value of $64xyz$ I tried adding $4xy +16yz + 8xz$ on all side this making it look like this $(x+2y+4z)^2 =...
2
votes
3answers
43 views

Intersection of 3 planes along a line

I have three planes: \begin{align*} \pi_1: x+y+z&=2\\ \pi_2: x+ay+2z&=3\\ \pi_3: x+a^2y+4z&=3+a \end{align*} I want to determine a such that the three planes intersect along a line. I do ...
0
votes
2answers
33 views

Easy mathematical transformation {equation} (still having problems)

so in my syllabus the professor wrote $q = 500 / p - 4$ from which he deduced $p = 500 / q + 4$ I cannot arrive at the same p, especially because of the positive 4...
0
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3answers
28 views

What product represents the solution to the system?

What product represents the solution to the system? $$-y+7x=14$$ $$-x+4y=1$$ I have that $y=\cfrac{7}{9}$ $x=\cfrac{19}{9}$ But how to place this as a setup of a product of two matrices?
4
votes
5answers
69 views

Find all pairs of intergers satisfying $x^2+11 = y^4 -xy$ and $y^2 + xy= 30 $

Find all pairs of intergers $(x,y)$ that satisfy the two following equations: $x^2+11 = y^4 -xy$ $y^2 + xy= 30 $ Here's what I did: $x^2+11 +(30) = y^4 -xy +(y^2 + xy)$ $x^2+41 = y^4 +y^...
1
vote
4answers
50 views

What is the easiest way to solve this? $x + \sqrt{y} = 11, \sqrt{x} + y = 7$ [closed]

$$x + \sqrt{y} = 11$$ $$\sqrt{x} + y = 7$$ What is the easiest way to solve this?
1
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2answers
48 views

Solving in Integer sequences

Each $x_n$ comes from the set $\{2,3,6\}$, these statements are true $x_1 + x_2 + x_3+\cdots+x_n = 633$ $\frac{1}{{x_1}^2} + \frac{1}{{x_2}^2} + \frac{1}{{x_3}^2}+\cdots+\frac{1}{{x_n}^2} = \frac{...
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votes
3answers
50 views

Find solution following the system of equations! [closed]

$abcde-a=357^{400}$ $abcde-b=359^{410}$ $abcde-c=361^{420}$ $abcde-d=363^{430}$ $abcde-e=365^{440}$ ($a,b,c,d,e$ are natural numbers) I don't have any idea. I just tried this, $$a(bcde-1)=357^{...
0
votes
2answers
29 views

Showing 2 simultaneous equations have a unique solution

Let a, b, c and d be real numbers that are not all zero. Let ax + by = p cx + dy = q be a pair of equations in the variables x and y with p, q ∈ R. Show this system of equations has a unique ...
0
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0answers
44 views

Difficult to understand a method for stiff ODE

I really didn't find a well explained step necessary to implement the (implicit Runge Kutta) Rosenbrock method ... I found this paper : here you can visulize eq. 9 but is not so clear, first of all ...
2
votes
1answer
43 views

Finding a general solution to homogeneous and nonhomogeneous systems

Given the nonhomogeneous system $\cases{ax+by+cz=d\\fx+gy+hz=k}$ with solutions $(2,-3,1)$ and the homogeneous system $\cases {ax+by+cz=0\\fx+gy+hz=0}$ with solutions $(-1,1,1),(1,0,1)$, ind the ...
-1
votes
1answer
45 views

Trying to evaluate $xz$ [closed]

$$x = 11-y$$ $$z = y-3$$ Evaluate $xz$ I've tried to multiply $$xz = 11-y(y-3)$$ However, there will be no exact solution from here. Regards
2
votes
2answers
132 views

solution to system of equations

In a system of equations AX = B. by Cramer's rule if det(A)=0, then the system will have no solution but on the other hand, in non-homogeneous system of equations suppose the rank of square matrix ...
1
vote
0answers
37 views

Solving system of super-ellipse equations

Super-ellipses are the equations that yield unit circles in different p-norms in $L^p$ space. I'm interested in the solution space in the unit square. My trouble is finding the solutions that are ...
1
vote
1answer
40 views

Number of Non negative integer solutions of $3a+2b+c+d=19$

Find Number of Non negative integer solutions of $3a+2b+c+d=19$ My attempt: we have $$2b+c+d=19-3a$$ Required solutions is coefficient of $t^{19-3a}$ in $$( 1-t^2)^{-1}(1-t)^{-1}(1-t)^{-1}=\frac{...
0
votes
3answers
58 views

How would you determine the value of $c$?

$$3a+2b = 4$$ $$\dfrac{1}{5a-2b} = \dfrac{2}{a+6b+5} = \dfrac{3}{c-4}$$ How would you determine the value of $c$? Regards!
0
votes
2answers
51 views

How does an equation that's not fully factored, end up outputting solutions?

In a textbook I stumbled across this: $(32-8a+2b)x+(32-6a+b)=0$ Thus, $32-8a+2b=0$ (1) , $32-6a+b=0$ (2) Now how exactly does that first equation, imply the 2 solutions? In this case the LHS ...
0
votes
1answer
29 views

A least-squares solution $\hat x$ of an inconsistent system $Ax=y$

I am having confusion with two statements: A least-squares solution $\hat x$ of an inconsistent system $Ax=y$ is a solution of the normal equations $A^TA x = A^Ty$, $\hat x$ can be found by reducing $...
2
votes
1answer
37 views

Find the line $y= x+\lambda d$ for $(n-1)\times n$ linear system of equations

Consider the equations $$\sum_{j=1}^n a_{ij}x_j = b_i, i = 1,\cdots, n-1$$ or equivalently, $Ax = b$ with $A\in \mathbb{R}^{(n-1)\times n}, b\in \mathbb{R}^{n-1}$ and $x\in \mathbb{R}^n$ ...
1
vote
1answer
21 views

interested in direction of solution of linear system

I have a system of linear equations $Ax = b$ where $x$ is unknown and $A, b$ are known. $A$ and $b$ are large, so there is a significant cost to compute the system In my particular applications, I ...
0
votes
2answers
55 views

How to solve these two equations for $\tau$ and $b$? All the other symbols are constants.

$$\large{\frac{\mu b}2\frac{2-\nu}{1-\nu}R\ln\frac{R}{r_c}-\tau\pi R^2+\frac{\gamma_0\pi^2 R^2}{b_0}\sin\left(\frac{2\pi(u_0+b)}{b_0}\right)=0}$$ $$\large{\frac{\mu b^2}4\frac{2-\nu}{1-\nu}\left[1+\...
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votes
1answer
42 views

It is known that the equation $ax + (b - 3)= (5a - 1)x+3b$ has more than one solution. Find the value of $100a + 4b$. [closed]

It is known that the equation $ax + (b - 3)= (5a - 1)x+3b$ has more than one solution. Find the value of $100a + 4b$.
2
votes
1answer
56 views

Elimination method algebra

I am having some trouble understanding this solution to a question which required the elimination method. It comes from page 266 of Head First Algebra. On the right hand side why does the + 0.4p ...
-2
votes
1answer
42 views

Find out The values for $a, b$ and $c$ [closed]

$6a+4b+9c = -1$, $4a+6b+9c= -1$ and $9a+9b+17c= 1$ I want to know about that how we'll find out the values of $a, b$ and $c$ from the above equations. The answers are $a=-3.25, b= -3.5$ and $c= 3.5$....
1
vote
3answers
92 views

symmetric polynomial recursion to solve the system, $x^5+y^5=33$, $x+y=3$

I was just reading on symmetric polynomials and was given the system of equations$$x^5+y^5=33 \text{ , } x+y=3$$ In the text they said to denote $\sigma_1=x+y$ and $\sigma_2=xy$, and to use recursion....
2
votes
1answer
44 views

Find the value of all $x$ satisfying $(f\circ g\circ g\circ f)(x)=(g\circ g\circ f)(x)$, where $(f\circ g)(x)=f(g(x))$.

Let $f(x)=x^2$ and $g(x)=\sin(x)\ \forall\ x\in \mathbb R$. Then find the value of all $x$ satisfying $(f\circ g\circ g\circ f)(x)=(g\circ g\circ f)(x)$, where $(f\circ g)(x)=f(g(x))$. Solution. ...
0
votes
1answer
18 views

Rank of a Matrix related to consistency

I have a question about the rank of matrices and its relation to a linear system being consistent. I do not intuitively understand why the rank of the coefficient matrix being equal to the rank of the ...
0
votes
0answers
24 views

Unconcrete polynomial value constraints

Let's say I have this general 3rd order polynomial defined: $f(x):=ax^3+bx^2+cx+d$ If I have concrete constraints of x and y values, I am able to solve the parameters. For example: $f(0)=0$ $f(1)=0$...
2
votes
0answers
43 views

Circular Motion - System of Differential Equations

The following system of differential equations describes a charged particle in a viscous medium enveloped in an EM field, $$\partial^2_t x(t) = a\cos(\omega t +\phi)y +bx +c\partial_t y -d\partial_t x$...
-1
votes
0answers
23 views

Complex set of linear equations

I'm trying to find $\beta$'s which solve the following problem: $\sum Vi \beta i = V$ Or that at least minimize ($\sum Vi \beta i - V)$ where $Vi$ are vectors. Additionally, there are a few other ...
0
votes
3answers
47 views

Using the Inverse matrix to solve a linear system.

I have the following question: $$x - 2y = 1$$ $$2x + 3y = 4$$ So let's say: $$A = \left[ \begin{matrix} 1 & -2 \\ 2 & 3 \end{matrix} \right]$$ So according to some formula to $2 \times ...
0
votes
1answer
36 views

Need to solve equations which take vector inputs. I have tried with matlab solve() but I'm getting zeros for both tau_new & b

rc=1; x=linspace(0,1,1000); R=exp(1).*x.*rc; b0=.3614/sqrt(2); eqn1=(b./(8*pi)).(1./R).(log(R./rc)+1)+(b./(2*b0))-tau_new==0, eqn2=(b./(2*pi)).(1./R).(log(R./rc))+(b./(b0))-tau_new==0]; vars=[...
0
votes
1answer
17 views

Inverse of trilinear interpolation: three degree-1 polynomials in three variables

I'd like to find the interpolation weights of a trilinear interpolation of a 3D-vector field. I'm given the vectors $x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x\in\mathbf{R}^3$ and I want to find the scalar ...