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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1answer
20 views

To solve the system of Diophantine equations.

I decided to compile a single task and to record such a system. $$\left\{\begin{aligned}&xt+yw=az^2\\&xw-yt=br^2\end{aligned}\right.$$ $a,b - $ integers that are the problem. It is clear ...
-3
votes
0answers
26 views

Ticket price mix, how many children and adults? [on hold]

Mega Movies hosted a film premiere on Friday night. They charged \$8 for adults and \$6 for children. One hundred six adults and children attended, and \$810 was made in ticket sales. How many ...
0
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0answers
9 views

How to solve this kind of difference equation?

How to find $v_k$, $k=0,1,2,\dots$ such that $$v_k + \sum_{n=1}^{k} \frac{\alpha^n}{n}v_{k-n} + \sum_{n=1}^{k}\frac{\beta^n}{n}v_{k+n} = 0,$$ where $\alpha,\beta \in \mathbb{C}$. ($v_i=0$ for ...
2
votes
1answer
12 views

Logic supporting column operations on matrices

In matrices, we justify row operations by drawing parallels with solving a system of equations i.e.: 1.Interchanging rows = Interchanging equations \ 2.Adding one multiple of a row to another = ...
-1
votes
1answer
21 views

find the distance between line and point R3 [on hold]

I would like to know how I can find the distance between the line and point in R3 the equations of line: $$ \left\{ \begin{array}{c} 2x+y+z=2 \\ 3x+4y-z=3 \end{array} \right. $$ and point (3, ...
0
votes
0answers
11 views

Is there an efficent way to solve large systems of purely quadratic equations?

I have the following system of quadratic equations $$ b_1 = \sum_{k=1}^R x_{i_1, k} \ y_{j_1, k} $$ $$ \vdots $$ $$ b_p = \sum_{k=1}^R x_{i_p, k} \ y_{j_p, k} $$ where $i_1, \ldots, i_p \in ...
0
votes
2answers
58 views

Matrix invertible iff det(matrix)$\neq 0$?

When we want to find the inverse of the matrix $$\begin{bmatrix}a & b \\ c & d\end{bmatrix}$$ we're searching for a matrix $$\begin{bmatrix}x & y \\ z & w\end{bmatrix}$$ such ...
1
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0answers
15 views

Solve $b_1 e^{-a_1x^2}-b_2 e^{-a_2x^2}-b_3 e^{-a_3x^2}=0, \forall x$

Suppose, \begin{align*} b_1 e^{-a_1x^2}-b_2 e^{-a_2x^2}-b_3 e^{-a_3x^2}=0, \forall x \end{align*} Assume $a_1,a_2,a_3, b_1,b_2, b_3>0$ What are the possible values of $a_1,a_2,a_3, b_1,b_2, ...
1
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3answers
60 views

Equation system modulo prime

I have an excercise, it is to solve $$9\equiv_{p}8k_1+k_2$$ $$32\equiv_{p}6k_1+k_2$$ $$45\equiv_{p}11k_1+k_2.$$ $k_2$ is easily eliminated from the equations but I don't know how to proceed from ...
0
votes
1answer
60 views

$S_{1}\iff S_{2}$ in complex numbers

Let : $a_0 , a_1 , a_2 , b_0 , b_1 , b_2 \in \mathbb{C} $ : Show the following equivalence : $$\begin{cases} ( 1 + a_0 ) ( 1 + a_1 ) ( 1 + a_2 ) &=& ( 1 + b_0 ) ( 1 + j b_0 ) ( 1 + j^2 b_0 ) ...
1
vote
1answer
27 views

Solution of overdetermined polynomial system

Some of you will find this question pretty straightforward to answer, but I desperately need some help in solving a problem involving several equations and 2 unknowns, for an engineering application. ...
-1
votes
2answers
27 views

how many jelly beans did each girl have at first?

Martha and Mary had $375$ jelly beans in all. After Mary ate $24$ jelly beans and Martha ate $\frac 17$ of her jelly beans, they each had the same number of jelly beans left. How many jelly beans did ...
0
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0answers
28 views

Quention about the historical definition of determinant

$$ax+by = k_1\\cx + dy = k_2$$ If I want to solve for $y$ in the first equation: $$by = k_1 - ax\implies y = \frac{k_1-ax}{b}$$ Then substitute $y$ in the second equation: $$cx + d\frac{k_1-ax}{b} ...
1
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3answers
82 views

Let $p^3+q^3=4$ and $pq=2/3$ . Find $p+q$.

Let $p^3+q^3=4$ and $pq=\frac{2}{3}$ . Find $p+q$. A graphing calculator can find values of $p$ and $q$ numerically. As one can see from the graph below, the two solutions are approximately ...
0
votes
0answers
23 views

Substitution in a system of ordinary differential equations when terms of the same order derivative for different variables occur in the same equation

Let's say I have a differential equation such as: y'' - 2ty' + y = 0, y(0) = 2.1, y'(0) = 1.0 I can solve this (among other ways) by substitution and conversion ...
1
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0answers
14 views

A system of non-linear equations with a small parameter

Is there any way to solve analytically the following system of equations to the leading order in $\epsilon$: $$\left\{ \begin{array}{rcl} \mu^2 \phi_1 + \lambda \phi_1 (\phi_1^2 + \phi_2^2) + ...
0
votes
2answers
58 views

Solving a system of three simultaneous equations

Given the system $$ \begin{align*} -2x + ay - bz &= -4 \\ x + bz &= 2 \\ 2x + y + 3bz &= b \end{align*} $$ The question asks to find conditions on $a$ and $b$ that the system has no ...
3
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4answers
157 views

How to solve current exponential equation? [closed]

There is an equation: $$3^x + 7^x = 21^x$$ How to solve this?
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0answers
16 views

How would I get Maple to display all integer solutions to this system of inequalities? [closed]

I need to find all the integer solutions satisfying: $$20+x\geq0;\space2x+5y\geq;\space-x-2y\geq0.$$ I'm not sure which Maple functions would work and whatnot. A guy can only google this stuff for so ...
1
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1answer
38 views

How can I solve a system of two equations, like $A + B = 13$ and $2D + B = 13$?

I am currently studying for my SSAT and this question appeared in my practice book: When $A + B = 13$ and $2D + B = 13$, what is the value of $D$? (A) 13 (B) 5 (C) -5 (D) -7 ...
1
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2answers
80 views

Solve system of 3 equations

$x+y+z=0$ $x^2+y^2+z^2=6ab$ $x^3+y^3+z^3=3(a^3+b^3)$ this is what i reasoned out so far; $xyz=a^3+b^3$ $x^2+zx+z^2=3ab$ $y^2+zy+z^2=3ab$ $x^2+xy+y^2=3ab$ $y^2=3ab+zx$ $x^2=3ab+zy$ ...
1
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1answer
28 views

Cannot figure out a second order lineary differential equation with initial values

I got the following question: Solve the following initial value problem: $y(0) = 0$, $y'(0) = 1$, $$y'' + 10y' + 25y = 0$$ So I started with getting the general solution: $$ y(x) = C_1e^{-5x} + ...
2
votes
0answers
25 views

$\sum (\sqrt{x_k-k^2}-k)^2=0$ implies $x_k=2k^2$?

Let $x_1,x_2,\ldots,x_n$ be reals numbers such that $$\sum_{k=1}^n k\sqrt{x_k-k^2}=\frac12\sum_{k=1}^n x_k$$ Find all possible $n$-tuples of solution. So, I got the following solution from ...
0
votes
3answers
25 views

How to solve this system of equations (Lagrange Multipliers)

I was doing a question on Lagrange multipliers and stucked when trying to evaluate the point. The system of equations that I can't solve is this: $$y^2-x^2+3x-3y=0$$ $$-y^2-yx+3y-xy=0$$ I just ...
1
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0answers
26 views

GMRES and Preconditioning

I am using GMRES to approximate the solution of a system of equations $Ax=b$, I am using a preconditioner $P$ to make GMRES converge faster. My question is how do I know if the preconditioner I am ...
4
votes
1answer
166 views

How to solve 29 coupled quadratic equations?

I have a set of 29 coupled quadratic equations, with 29 unknown variables. Can anyone offer any advice on how I could go about solving this? 3 days of staring at a wall has so far given me no ...
1
vote
1answer
29 views

What's the solution to this exponential system of equation?

What are the steps to solving a system of equations when $x$ and $y$ are exponents? But they have different base. Here is the problem. $5^x\times3^y=45$ $3^x\times5^y=75$
1
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4answers
69 views

Solving a homogeneous system of three ODEs with variable coefficients.

I am interested in solving the following system of ODEs: $$ \begin{pmatrix} x'(t) \\ y'(t) \\z'(t) \end{pmatrix} = a \begin{pmatrix} 0 & -B_2 & B_1 \sin \omega t \\ B_2 & 0& -B_1 ...
1
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1answer
63 views

Solving polynomial equation system to find three dimensional location

For an embedded systems project, I need to solve a system of equations. However, my algebraic skills are limited, and I am not able to solve it. This question consists of the following parts. The ...
2
votes
2answers
17 views

Finding the nth term in a recursive coupled equation.

I'm probably missing something simple, but if I have the recursive sequence: $$ a_{i+1} = \delta a_i+\lambda_1 b_i $$ $$ b_{i+1} = \lambda_2 a_i + \delta b_i $$ how would I find a formula for $a_n$, ...
2
votes
0answers
29 views

Why are equilibria so important?

In studying nonlinear systems of differential equations, unlike linear systems, it turns out that we are more interested in equilibrium points rather than general solutions themselves. I mean, look ...
4
votes
1answer
31 views

linearly independent and determinant

This question says a matrix $\begin{bmatrix}a & b\\c & d\end{bmatrix}$ where $a_{ij}$ are real numbers. I need to prove that $\det|A|=ad-bc\neq0 \iff $the columns are linearly independent. ...
1
vote
1answer
33 views

Solving the equations $x_1= 4 x_2$ and $x_3= 5 x_2$, with the sum of all three being $150$

Here is the problem. A set X is partitioned into subsets x1, x2, and x3. The number of elements in x1 is 4 times the number in x2. And the number in x3 is 5 times the number in x2. If n(x)=150, ...
2
votes
0answers
38 views

Impossible System of Equations

This is from a competition: DMM Olympiad, Ural State University P4 I don't understand what the question means exactly (the first part, i.e. "exclude $x$ or $y$ from..." part). Does it mean "write $x$ ...
1
vote
2answers
51 views

How to solve this Linear Algebra problem involving a system of linear equations?

The following is what I have so far. I'm not sure how to use my echelon matrix to find out which values for the variables can provide an answer to the question or how to prove it. I was thinking ...
0
votes
1answer
22 views

Help establishing restrictions for consistency on a linear system.

I'm having trouble wrapping my head around this problems, and others similar to it. I can typically solve systems of linear equations, but some give me trouble, especially dealing with unknown ...
0
votes
0answers
25 views

Search Direction in Conjugate Gradient

Could you help me with a Conjugate Gradient question? In using CG to solve $Ax = b$, why is the search direction $p_{k+1}$ in CG chosen as a linear combination of the residual $r_k$ and previous ...
1
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1answer
25 views

A Set of Linear Equations Equal to Zero

If A+B+C+D = 1; Let A,B,C,D be elements of vector V. Let there be a 4*4 matrix M as such : -5 5 0 0 4 -8 4 0 0 5 -7 2 0 0 10 -10 A property ...
0
votes
2answers
41 views

Find Maximum and Minimum value by two polynomial equations

Suppose there are $7$ real numbers say $A,B,C,D,E,F,G$ All we need to find the minimum and maximum value of $G$ satisfying the following two equations :- Sum of Numbers :- $A + B + C + D + E + F ...
2
votes
2answers
56 views

Find a, b, c if three equations are given?

I was given three equations in term of $a, b$ and $c$. Equations are as follows $ab (a+b+c)=1001$ $bc(a+b+c)=2002$ $ac(a+b+c)=3003$ Find $a, b, c$. MY ATTEMPT I took tue ratio and I got relation as ...
1
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0answers
13 views

independent nonlinear equations

I am just wondering what the term 'independent equations' means. I found the term in the book of Kolmogorov about the basic notions of probability calculus. After Definition I of Section ...
0
votes
1answer
22 views

How get variable values when one of them is within non-trivial radical

Can this be done, and if so, how: Given four constants $(K_i)$ and variables $x$ and $y$, is there a way for me to find the values of $x$ and $y$ when $K_{1-4}$ are known for these two simultaneous ...
2
votes
2answers
53 views

How to prove, that solution of system $Ax = b$ exists only if there is no solution of $A^T y = 0$ and $b^T y = 1$?

I have a little linear algebra problem here: How can I prove, that there is a solution of system $Ax = b$ only if there is no solution of $A^T y = 0$ and $b^T y = 1$?
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0answers
19 views

Trouble finding the values of a matrix using rref

I'm working on a school project in which I have to get all the values of a missing matrix. To make a small test in my program, I used a simplified example just to see if the math was right, but for ...
4
votes
1answer
68 views

Find minimum number of sums of $3$ out of $5$ whose vanishing implies all five to be zero

Problem: let $a,b,c,d,e$ be real numbers, now there are $\left(\binom{5}{3}=10\right)$ numbers $$a+b+c,a+b+d,a+b+e,a+c+d,a+c+e,a+d+e,b+c+d,b+c+e,b+d+e,c+d+e$$ Question1:($\textbf{Jérémy Blanc ...
0
votes
1answer
23 views

Determinants to solve a system

I was reading a book on Calculus when I came across this: $$\begin{cases} v+\ln(u)=xy \\ u+\ln(v)=x-y \\ \end{cases}$$ $$\begin{cases} \frac1u\frac{\partial u}{\partial x} +\frac{\partial ...
2
votes
0answers
31 views

suggestion for lyapunov function

Consider differential equation \begin{align}x'&=-t(x+y)\\ y'&=-y+x-y(y^2-6).\end{align} Can some one suggest a lyapunov function for it. I have examined $V(x,y)=x^2+y^2$ , ...
0
votes
1answer
21 views

Solve for minimum x value in two-variable equation provided a ratio

The equation is: $$500 = 5x - 2y$$ I know the ratio of x to y is $7:3$; what is the minimum possible value of x and how would I approach this question?
0
votes
1answer
22 views

Understanding Overdetermined System

Consider a system of linear equations $$A \times x = B$$ The system has a unique solution exactly when the determinant of the coefficient matrix (i.e. A) is nonzero. When the determinant of the ...
0
votes
0answers
24 views

System of Trigonometric Equations

Could someone please help me with the following system of equations