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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2
votes
1answer
19 views

system of equations with $n$ equations and $2^k n$ unknowns

I have a system of equations with infinitely many solutions. I would like to find a "nice" way to write down an explicit solution. Here, $n,k\geq 1$ are integers, we have $x_1,x_2,\dots, x_{2^k n}$ ...
0
votes
2answers
29 views

Explain how solution got $c_1$ and $c_2$

Can someone explain how the solution manual got $c_1$ and $c_2$ in this:
0
votes
0answers
20 views

Solving a system of differential-algebraic equations

I am seeking for references around the topic of solving this type of differential equation system : $$ \left\{ \begin{array}{ll} \partial_x y_{i+1}(x) = y_{i+1}(x)-y_{i}(x) \\ y_{i+1}(x) = y_i(f(x)) ...
1
vote
0answers
28 views

Convert a 2D autonomous ODE system into a 1D system?

Suppose I have two equations: $$\frac{dx}{dt} = x(2-x-y),\, \frac{dy}{dt} = ky(2-ax-by)$$ that together form a 2D ODE system ($x$ plotted against $y). K, a$ and $b$ are all independent positive ...
0
votes
0answers
27 views

Conormal Points Parabola

Let the line $lx+my=1$ cut the parabola at $y^2=4ax$ in the points A and B.Normals at A and B meet at a point C. Normal from C other than these two meet at D.Then coordinates of D are? I tried to ...
0
votes
0answers
28 views

Systems of equations of the form $\sum_{i \in I} \sum_{j \in J_i} v_i \times v_j = a$

Is there any theory that deals (directly or not) with systems of equations of the form $$\sum_{i \in I} \sum_{j \in J_i} v_i \times v_j = a,$$ where $a \in \mathbb{R}^3$ is known, $v_i, v_j \in ...
0
votes
1answer
34 views

Optimizing for the minimum relative distance in a given situation?

I have primarily been working on this problem for quite some time now; the level of the problem is introductory calculus w/ optimization problems. The situation is as follows: Ship A sails due ...
0
votes
0answers
22 views

Ordinary differential equation system with maximum value of x+y

I've got the following simple system of ODEs: x': x*(1-x-y) y': y*(1-a* y-b* x). Plotting the system in a x' y' plot is not fully satisfying to me as what I want to achieve is that the sum of x and ...
5
votes
2answers
89 views

How to prove whether the equation set has a unique solution?

\begin{eqnarray} \begin{cases} \sin A \sin C-(\sin B)^2=0 \cr AC-B^2=0 \cr A+B+C-\pi=0 \cr A>0,B>0,C>0 \end{cases} \end{eqnarray} How to prove whether the equation set has a unique solution ...
1
vote
3answers
25 views

multiplying term on sum

Say I know the following relation holds $$ \sum_i f_i + \sum_i g_i = 0 $$ Now I multipy both sides with a set of vectors $\mathbf v_i$. Will it still be true that $$ \sum_i f_i \mathbf v_i + \sum_i ...
1
vote
2answers
39 views

Solving Systems of Linear Differential Equations by Elimination

For a homework problem, we are provided: $\frac{dx}{dt}=-y + t$ $\frac{dy}{dt}=x-t$ Putting these into differential operator notation and separating the dependent variables from the independent: ...
1
vote
0answers
19 views

I need help with creating linear equations from multiple points on a graph

How do you create a linear equation from multiple points on a graph I am working on a question where the points are $(-3,8),(2,5)$ and $(7,2)$ and I need to find out how to create a linear inequality ...
1
vote
1answer
26 views

solution of a system of equation

Let $A\in M_{m\times n}(\Bbb R)$ and let $b_0\in \mathbb R^m$. Suppose that the system of equations $Ax=b_0$ has a unique solution. Which of the following is true? $Ax=b$ has a solution for every $b ...
1
vote
2answers
120 views

Solve a system of two nonlinear equations

$$ \begin{cases} x^2 - y^2 + 12y - 21 = 0\\ 2x^2 + y^2 + 2xy + x = 0 \end{cases} $$ I've tried the change of variables: $u = x + y$, $v = x - y$ After it I've got: $$ \begin{cases} uv + 12\frac{u - ...
2
votes
1answer
46 views

A system of linear equations with 3 variables such that its solution set is: $\{(a,b,c)|a^2=b \}$?

Is there a system of linear equations with 3 variables such that its solution set is: $\{(a,b,c)|a^2=b \}$? It's enough to show that for one equation: $Ax+By+Cz=D$ the solution set doesn't work. ...
1
vote
2answers
35 views

Solve the following in vector form:

So i did a substitution to solve the system normally, and got $x=17.67$ $y=9.67$ $z=10.67$ Where I am stuck is how to represent something like this in a vector form, maybe my solution was wrong ...
1
vote
1answer
39 views

Rearranging An Equation To Solve?Can't?

How would I rearrange the equation: $$a=b^{(c/d)}$$ to find c?
1
vote
0answers
38 views

least square solution of overdetermined system with additional unknown

I was hoping somebody could tell me the best way to solve the following overdetermined system for the scalars $x_{1}$,$x_{2}$ and $x_{3}$, where the C $3 \times 1$ vectors are unknown, $A_{i}$ is a $3 ...
1
vote
0answers
23 views

canonical form of parabolic-type PDE involving exp(x) and ln(x)

The attached picture skips a lot of the work, but I've worked this problem at least 6 times in the last 8 hours, still getting stuck at reducing to canonical form - that is, trying to solve for x and ...
0
votes
2answers
31 views

3 Variable System of Equations When All Set to Zero

So I'm doing an a bit of a pre-assessment for something, and I feel like I am missing something on this question: Now I know how to solve a normal 3 variable system, but with this they are all set ...
1
vote
1answer
49 views

Analytic solution of a system of four second order polynomials

Can I systematically solve in $\mathbb{R}^4$ the following system without using Grobner basis algorithm ? If not, can I find the exact number of solutions ? $$ \begin{equation*} \left\{ ...
0
votes
1answer
26 views

Solution for associated homogeneous linear system

If an inhomogeneous system of linear equations has an associated homogeneous system that has only the trivial solution, then how can I show that the inhomogeneous system has exactly one solution?
6
votes
1answer
156 views

Amount of solutions to the Diophantine equation of Frobenius

The Diophantine equation of Frobenius is any equation of the form: $$\sum_{i=1}^k a_i x_i = n$$ where the $a_i$'s are given and so are $k$ and $n$. I'm looking for an algorithm to compute the number ...
0
votes
1answer
60 views

Is it possible to create any combination of areas?

Given a point $P(x,y)$ in the unit square, two polygons Blue and Green are defined by drawing a 45-degree line through $P$ and creating polygons with the top-left and bottom-right corners, ...
1
vote
1answer
27 views

Behaviour of roots of a polynomial with function coefficients

Let $(-1+c_4(h))x^4 +c_3(h)x^3+c_2(h)x^2+c_1(h)x+c_0(h)=0$ be an equation with variable coefficients, depending smoothly on $h$. Also let $0\le c_4(h)\le 1-\epsilon$ for some $\epsilon>0$ and ...
1
vote
3answers
37 views

gaussian elimination to solve a question (using a paramter)

I want to solve : x2+x3=0 -x1 -x3=0 x1-x2 =0 I got the $x_1 = -t, x_2=-t, x_3=t$. But the book has $x_1 = t, x2=t, x3=-t$. ...
0
votes
2answers
29 views

Find system of equations such that

Find system of equations that will describe: a) plane $M \subset \mathbb{R}^3$ passing through the points $(6,1,-3), (1,5,1), (1,8,2)$ b) line $L \subset \mathbb{R}^3$ passing through $(1,2,-1), ...
1
vote
2answers
66 views

Solving a non-linear, multivariable system of equations

I'm researching the mathematics behind GPS, and at the moment I'm trying to get my head around how to solve the following system of equations: $\sqrt{(x-x_1)^2+(y-y_1)^2+(z-z_1)^2}=r_1$ ...
0
votes
1answer
49 views

Solving 4 unknowns with 4 equations all equal to zero

I have the following equations: $$\begin{align} a&=0.35a+0.35d\\ b&=0.65a+0.65d\\ c&=0.35b+0.35c\\ d&=0.65b+0.65c \end{align}$$ I know $b=d$, but where do I go from here? I have a ...
0
votes
2answers
48 views

Solve 5 unknowns with 5 equations

I have the following set of equations: $A = x_1x_2x_3x_4x_5$ $B = x_1x_3x_5 + x_1x_4x_5 + x_2x_4x_5 + x_2x_3x_4$ $C = x_3 + x_4 + x_5$ $D = x_1x_2x_3x_4$ $E = x_1x_3 + x_1x_4+x_2x_4$ Where A, B, ...
0
votes
1answer
62 views

How to solve the system of 4 equations of four unknowns

Solve this system of the four equations of four unknowns $a, b, c, d>0 $ $$ 165(a+b+c)=abc\tag1 $$ $$220(a+b+d)=abd \tag2 $$ $$297(a+c+d)=acd\tag3 $$ $$540(b+c+d)=bcd \tag4 $$ I tried to ...
3
votes
2answers
35 views

How to solve the given system of equations for $I_1, I_2, I_3$?

I have this system of equations: \begin{cases} I_1 = I_2 + I_3 \\ \epsilon_1 - I_1(R_1 + R_2) - I_2 R_3 = 0 \\ \epsilon_1 - I_1(R_1 + R_2) - I_3(R_4 + R_5) + \epsilon_2 = 0 \end{cases} I want to solve ...
0
votes
0answers
39 views

A simple non-linear equation.

Let ${\rm a}$ and ${\rm b}$ be two given vectors in ${\mathbb R}^n$. Find ${\rm u}\in {\mathbb R}^n$ and $x\in[0,\infty)$ such that $$ {\rm a} +{\rm b}x+\frac{1}{2}x^2{\rm u}=0, \quad \|{\rm u}\|=1 ...
0
votes
0answers
28 views

How can I find multiple solutions for a system of equations?

I'm writing a program for CheckIO.org that is supposed to return an array, $$ \begin{bmatrix} x\\ y\\ z \end {bmatrix} $$ , that satisfies the System of Equations $$ A \begin{bmatrix} x\\ y\\ z \end ...
2
votes
0answers
18 views

Solve Intergal Equation of form g.u1=Int(K.u2) for u1 and u2

I'm trying to find a solution to a differential equation of an unusual form: $$g(x) u_1(x)=\int_a^b K(x,y) u_2(y) dy$$ where $g(x)$ and $K(x,y)$ are known and $u_1(x)$ and $u_2(x)$ are complex ...
1
vote
1answer
36 views

Find integral of the 2 by 2 system of ODE

We want to find a function $F(x(t),y(t))=c$ where $x(t),y(t)$ are solutions to the system $\begin{bmatrix} \dot x=\frac{t-y}{y-x} \\\dot y=\frac{x-t}{y-x}\end{bmatrix}$. Such a function $F(x(t),y(t))$ ...
2
votes
1answer
24 views

Add together simple equations

I have three equations $$ x = 20 \\ -x+a = 10 \\ y = 2 $$ Can I add these equations and get $$ x-x+a+y = 20+10+2 \\ a+y = 32? $$ If yes, what is the name of the rule applied?
2
votes
0answers
20 views

Solving equation set with boolean operators and very specific format

I have to write a program to solve a set of equations like the following (+ is XOR and * is ...
0
votes
0answers
13 views

Show the following system is not possible

Assume throughout that the base field is the prime field $\mathbb{F}_2$. I have two $n \times n$ matrices: $I_n$, the $n \times n$ identity matrix, and $C_n$ the matrix obtained from $I_n$ by shifting ...
0
votes
1answer
28 views

System of homogeneous linear equations issue

Solve the following system: $4x-12y+z=0\\ x-5y-z=0\\ -4x+12y+z=0$ So in matrix form it is $ \left(\begin {matrix} 4 & -12 & 1 \\ 1 & -5 & -1 \\ -4 ...
1
vote
1answer
23 views

Find base vectors and dim

Find base vectors and dim of a space described by the following system of equation: $$2x_1-x_2+x_3-x_4=0 \\ x_1+2x_2+x_3+2x_4=0 \\ 3x_1+x_2+2x_3+x_4=0$$ I did rref of the matrix and as a result i get: ...
1
vote
1answer
45 views

How can I solve the following exercise

How can I solve the following exercise $$φ_1(x)=e^x-\int_{0}^{x}φ_1(t)dt+4\int_{0}^{x}e^{x-t}φ_2(t)dt$$ $$φ_2(x)=1-\int_{0}^{x}e^{-x+t}φ_1(t)dt+\int_{0}^{x}φ_2(t)dt$$
0
votes
0answers
30 views

What are “symmetry arguments” in the context of solving systems of equations?

What and how are the "symmetry arguments" used to solve a system of equations? My text makes extensive use of this argument but do not provide and explanation of how it works or the definition of ...
0
votes
1answer
24 views

Solve $|x - z_1| = d_1 + y$ and $|x - z_2| = d_2 + y$ simultaneously for $x$ and $y$

Given the two equations $|x - z_1| = d_1 + y$ and $|x - z_2| = d_2 + y$ , and suppose that $z_1, z_2 \in \mathbb{R}$, $z_1 \neq z_2$ and $d_1, d_2, \in \mathbb{R}_{> 0}$ are all known reals, solve ...
2
votes
1answer
41 views

System of equations, 3 equations and 3 unknown

I have a system of equations that I'm trying to solve, $Mb = x$ $M=\begin{bmatrix} e^z &e^z &e^z \\ aX_1 &bX_1 &cX_1 \\ aX_2 &bX_2 &cX_2 \end{bmatrix}$ $b = ...
0
votes
1answer
40 views

Gauss' method to solve the following system of equations

I need to use Gauss' method to solve the following system of equations and to describe its solution set. Can anyone help me getting started. \begin{alignat*}{8} x & + & y & + & z ...
-1
votes
2answers
40 views

Solve three simultaneous equations with 3 unknowns

(b) An electrical circuit comprises three closed loops giving the following equations for the currents $i_1, i_2$ and $i_3$ \begin{align*} i_1 + 8i_2 + 3i_3 &= -31\\ 3i_1 - 2i_2 + i_3 ...
-1
votes
1answer
61 views

Does $x^2=83\pmod{101}$ have solutions? without calculating them

Does $x^2=83\pmod{101}$ have solutions? without calculating them. I'm not sure how to tackle this without solving, I tried using chinese remainder and quadratic reciprocity.
0
votes
2answers
79 views

Solve a differential equation system

I am solving a physics problem and done almost everything to a couple of last steps. I currently have two differential equations which I need to solve for functions$x(t)$ and $y(t)$: $$ \ddot x =\dot ...
1
vote
0answers
31 views

Solutions to linear equations

Am I right in thinking that the following augmented matrix equation only has one solution: $\begin{bmatrix} 0 & 1 & 0 & 4\\ 0 & 0 & 1 & 10 \end{bmatrix} $ i.e., if the ...