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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1
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1answer
30 views

Let $A$ be an $m \times n$ real matrix and $b \in \Bbb R^m$ with $b \neq 0$ . Then

The set of all real solutions of $Ax=b$ is a vector space. If $u$ and $v$ are two solutions of $Ax=b$, then $\lambda u+ (1-\lambda)v$ is also a solution of $Ax=b$ $\forall \lambda \in \Bbb R$. ...
2
votes
1answer
68 views

Specific system of differential equations

I have the following system of equations: \begin{eqnarray}\frac{dx}{dt} = x(1 - x^2 - y^2) \\ \frac{dy}{dt} = y(4 - x^2 - y^2) \end{eqnarray} I want to prove that if a solutions starts (at time $t = ...
1
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4answers
65 views

Simultaneous Equations (Stuck on the algebra)

Question: Solve the following simultaneous equations for real values of x and y $$ \left\{ \begin{array}{l} 9^{2x+y} - 9^x \times 3^y = 6 \\ \log_{x+1}(y+3) + \log_{x+1}(y+x+4) = 3 ...
4
votes
1answer
60 views

How to find out the value of $n$ in the given expression.

How can I find out the given expressions value of n? $$\frac{a+b}{2}=\frac{a^n+b^n}{a^{n-1}+b^{n-1}}$$ I tried multiplying both sides by denominator, but it didn't work. Also, observing tells me one ...
1
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3answers
25 views

Find the area of the region determined by the system: \begin{align} y & \ge |x| \\ y & \le -|x+1| +4 \\ \end{align}

Find the area of the region determined by the system: \begin{align} y & \ge |x| \\ y & \le -|x+1| +4 \\ \end{align} My attempt Assuming $x>0$ I have the system ...
0
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0answers
35 views

How to solve these equations with 4 equations and 3 variables?

Let $X, Y,$ and $Z$ be the variables of interest. $$X = Y + Z + C_1 \\ Y = C_2X \\ Z = -C_3 \\X = f(C_4)$$ They are presented in an economics book as an exercise, and I can't make sense of it. If $X$ ...
0
votes
1answer
56 views

Linear approximation of a stable manifold

Given $$ \begin{cases} \dot{x} = -x + y^2\\ \dot{y} = x - 2y +y^2 \end{cases} $$ Find a linear approximation of the STABLE manifold for the equilibrium $(1,1)$. My attempt: By using the Principle ...
3
votes
2answers
59 views

Find all $(x,y)$ satisfying $(\sin^2x+\frac{1}{\sin^2 x})^2+(\cos^2x+\frac{1}{\cos^2 x})^2=12+\frac{1}{2}\sin y$

Find all pairs $(x,y)$ of real numbers that satisfy the equation $(\sin^2x+\frac{1}{\sin^2 x})^2+(\cos^2x+\frac{1}{\cos^2 x})^2=12+\frac{1}{2}\sin y$ I supposed $a=\sin^2x$ and $b=\cos^2x$ So the ...
3
votes
2answers
52 views

Find $a,b$ for which $xyz+z=a,\quad xyz^2+z=b,\quad x^2+y^2+z^2=4$ has unique solution

Find all values of $a,b$ for which the system of equations $xyz+z=a,\quad xyz^2+z=b,\quad x^2+y^2+z^2=4$ has only one real solution. $xyz+z=a$ $xyz^2+z=b$ So,$\frac{xy+1}{xyz+1}=\frac{a}{b}$ I can ...
3
votes
1answer
92 views

Periodical solutions of this system of differential equations

We have the system of differential equations: $$x'=(1+m)y+x(1-(x^2+y^2))(4-(x^2+y^2)),$$ $$y'=-x+y(1-(x^2+y^2))(4-(x^2+y^2)),$$ with $m>0$. How do I show that $(0,0)$ is the only (instable) ...
1
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1answer
27 views

Find critical points of an ODE system

Find the critical points and the solution of the ODE system. Parameters $k_t, \space k_i $ are both positive constants. $$\frac{d \vec u}{dt} = k_t \begin{bmatrix}-1 & 1 \\1 & -1 ...
2
votes
3answers
66 views

Basic question on the infinitely many solutions of a linear system Ax=b,

I just want to verify the geometry of solutions to $Ax=b$, for the case when we have infinitely many solutions: If say for a $3\times 3$ matrix, after Gaussian Elimination, I have two pivot variables ...
0
votes
2answers
124 views

Solve system of equations $3|x|+2y=1, 2x-|y|=4$ [closed]

help with this system of equations \begin{cases} 3|x|+2y=1 \\ 2x-|y|=4 \\ \end{cases} I have no idea.
0
votes
1answer
23 views

Equation about polynomials

Is it possible to find real numbers $x_0, x_1, A_0$ and $A_1$ such equation : $$ (25a+32b+45c+80d) = A_0(ax_0^3 + bx_0^2 + cx_0 + d) + A_1(ax_1^3 + bx_1^2 + cx_1 + d) $$ is true for every real ...
0
votes
1answer
38 views

Solving a system of quadratic equations which evaluates to a 4th grade equation

I have to solve the following system of equations: $x^2 + 4y + 2 = 22$ $2y^2 + x + 6 = 40$ I tried to solve for one variable and then substitute it into the other equation, but a problem appears: ...
0
votes
1answer
35 views

Solve this system of equations explicitly for $f_d$ and $\alpha$

In this system of equations $\alpha$ and $f_d$ are unknown. $\alpha$ and $C_{13}$ are complex numbers; the other parameters are all real numbers. I want to solve this system explicitly for $f_d$ ...
1
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2answers
19 views

Graphing Parabolas ( finding area)

Find the area of the triangle formed by the origin and the points of intersection of parabolas $y=−3x^2+20$ and $y=x^2−16$. I tried graphing it. I could figure out what it meant when it's said formed ...
1
vote
1answer
64 views

Solve a system of polynomial equations

I cannot find a way to solve the following polynomial equation system $$\begin{cases}-3x^{2}+y^{2}-2xy-\alpha_{1}x+\alpha_{0}=0\\ -x^{2}+3y^{2}+2xy-\beta_{1}y+\beta_{0}=0\end{cases}$$ Could you help ...
2
votes
3answers
67 views

How to solve for angle for simultaneous “additive trigonometric” equation

I've been trying to study concepts from the field of inverse-kinematics, but have run into a mathematical roadblock. To solve for an angle given a number is quite simple in itself $$ \sin(\theta) = ...
5
votes
0answers
45 views

Why does the taylor expansion of a nonlinear system of differential equations exist if it has continuous second order partial derivatives?

My textbook states that for a nonlinear autonomous system $$x^\prime = F(x,y)\qquad y^\prime = G(x,y)$$ The system is locally linear in the neighborhood of a critical point $(x_0,y_0)$ whenever ...
1
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2answers
37 views

System of linear equations with whole numbers

I need some advice regarding this question. "Show that if in the system $Ax=b$, the determinant of $A$ is equal to $-1$, and all the members of $A$ are whole numbers, and all the members of $b$ are ...
0
votes
1answer
39 views

Number of solutions for system of elementary symmetric functions?

The elementary symmetric polynomials appear when we expand a linear factorization of a monic polynomial: we have the identity $$ \prod _{j=1}^{n}(\lambda -X_{j})=\lambda ...
3
votes
4answers
82 views

Solving Symmetrical Equations Algebraically

I'm doing some Cambridge STEP papers and have come across a tricky set of equations. \begin{align*} 99 &= c^3 + 6 cd^2 \tag{1} \\ 70 &= 3c^2d + 2d^3 \tag{2} \end{align*} From looking ...
3
votes
2answers
35 views

Finding the value of an expression given three equations.

$$x+y+2z=22$$ $$3x-2y+z=6$$ $$7x+3y-5z=1$$ above are three equations and we have to find the value of $$x^2+y^2+z^2=?$$
0
votes
1answer
67 views

Simultaneous equations in polar coordinates

I want to find the intersections of pairs of curves in polar coodinates. As an example, I have three circles with different offsets in a plane which you can see here. The offsets are: ...
3
votes
2answers
132 views

System of quadratic equations over field of size 2

I am working on system of quadratic equations. \begin{cases} (\alpha_1^1x_1+\ldots+\alpha_n^1x_n)(\beta_1^1y_1+\ldots+\beta_m^1y_m)=0\\ \ldots \\ ...
0
votes
2answers
36 views

What does “multiple non-trivial solutions exists mean?”

I came across this question of solving system of linear non-homogeneous equations : $$x+2y+z+4w=2$$ $$3x+6y+3z+12w=6$$ Options are : Only the trivial solution $x=y=z=w=0$ exists There is no ...
1
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0answers
23 views

If $F(t,y)$ is continuous in $\Bbb \Omega$ and linear in the variable $y$, does it follow that $F$ is Lipschitz on $y$?

I'm having this doubt: If $F(t,y)$ is continuous in $\Bbb \Omega$ and linear in the variable $y$, does it follow that $F$ is Lipschitz on $y$? This arose from the following full problem ...
1
vote
3answers
38 views

Which coefficients make the matrix $A$ singular

I am working on this exercise: Which numbers a, b make the matrix singular? the matrix is the following: $\begin{bmatrix}1 & 2 & 0\\a & 8 & 3\\0 & b & 5\end{bmatrix}$ I ...
0
votes
3answers
61 views

Systems of 2 non linear equations (2 variables)

I have a problem finding solutions to systems that contain 2 equations with 2 variables each and are second degree. Example : \begin{cases}x^2 + x + y^2 + y - 18 = 0 &&(1)\\ x^2 + xy + y^2 ...
3
votes
4answers
72 views

show that $a_1+a_2+a_3+a_4=8$ and that $64a_1+27a_2+8a_3+a_4=729$ given the following

Consider the sistem of equations: $$\begin{cases} a_1+8a_2+27a_3+64a_4=1 \\ 8a_1+27a_2+64a_3+125a_4=27 \\ 27a_1+64a_2+125a_3+216a_4=125\\ 64a_1+125a_2+216a_3+343a_4=343\\ \end{cases} $$ These ...
4
votes
2answers
50 views

Prove that it is not possible to assign the integers $1,2,3,\cdots,20$ to the twenty vertices of a dodecahedron so that each face have constant sum

Prove that it is not possible to assign the integers $1,2,3,\cdots,20$ to the twenty vertices of a regular dodecahedron so that the five numbers at the vertices of each of the twelve pentagonal ...
2
votes
2answers
102 views

Systems of Equations - calculate avg $x$ & $y$ given $x$ & $y$ are normally distributed

Given a number of equations (say, 30), and assuming x & y are normally distributed, how would I go about determining the average value of x & y? $10x_{1} + 60y_{1} = 5900$ $20x_{2} + 80y_{2} ...
1
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2answers
40 views

If $ 3x^2 + 2\alpha xy + 2y^2 + 2ax - 4y + 1 $ can be resolved into two linear factors, then prove the following.

Prove that : $ \alpha $ is a root of the equation $ x^2 + 4ax + 2a^2 + 6 = 0 $. What does it mean by "can be resolved into two linear factors"? If it means $( ax + b ) ( cx + d )$ , is it necessary ...
0
votes
0answers
40 views

closed form solution of the following iterative equation?

is it possible to obtain a closed-form solution w.r.t. ${P_j:\forall j}$ (or in terms of special functions) for the following equations: $\frac{\lambda}{\mu}P_0=P_1$ ...
2
votes
3answers
54 views

Solving a simultaneous equation

How can I solve the following simultaneous equations: $$3x^4+3x^2y^2-6xy = 0\tag 1$$ $$-2x^3y+3x^2-y^2=0\tag 2$$ I have tried rearranging for $y$ in eq(1) and plugging it into eq(2), but the result ...
1
vote
1answer
30 views

Let $a,b,m,n \in N$ with $\gcd(m,n)=1$ prove that the modular system $ \{ x=a \mod m ; x =b\mod n \}$ has absolution and is unique modulo $mn$}

Let $a,b,m,n \in N$ with $\gcd(m,n)=1$ prove that the modular system $ \{ x=a \mod m ; x =b\mod n \}$ has absolution and is unique modulo $mn$} Note that had asked a question I got why there it is ...
2
votes
0answers
50 views

Are two linear system equivalent?

Let $A$ and $M$ be square matrices of size $s$ and $n$ respectively, let $k_i \in\mathbb{R^n}$ be column vectors for all $i=1,\ldots,s$. Denote $K=\left[ \begin{matrix} {{k}_{1}} \\ \vdots ...
0
votes
0answers
13 views

Simultaneous linear congruence

I'm following a sample solution from my lecturer and part of it is to solve $x \equiv 9 (mod 12) $ and $x \equiv 3 (mod7) $ simultaneously. He writes it as $9 + 12k \equiv 3 (mod 7)$ which I ...
3
votes
2answers
40 views

Absolute value equation with rational expression

I am to solve the equation: $|\frac{2x}{x^2 - 3} | < 1$ And so: 1. I rewrote it as $|2x| < |(x - \sqrt 3) | |(x + \sqrt 3)|$ And I tried to divide it into a few intervals For $ ...
0
votes
0answers
35 views

Solve a matrix equation..

I am at moment trying to solve a system of linear equations, and I am not sure if the value I retrieve is even possible, or my program returns some garbage value... The equation I am trying to solve ...
0
votes
1answer
40 views

Solving a system (3) of nonlinear equations

I want to solve the following euqations: $6y-2xz = 0 $ $6x-2yz = 0 $ $x^2 + y^2 -8 = 0 $ One way would be to multiply the first equations with y and the second with x and then substract the first ...
0
votes
1answer
55 views

System of differential equations with multiple eigenvalues and one repeated

I need help with system of differential equations. $$x'(t) = Ax(t)$$ where $$\begin{align} A = \begin{bmatrix} 1 &-1& 2& 0 \\ -1& 1& 0& 0 \\ -1 &0 &1 &0 \\ 0 ...
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votes
1answer
25 views

Exact Equations and Integrating Factor

I am really having difficulty with this problem. I understand part a, as I did d/dx[x^2y'+(x^2'-2x)y]=0 and then took the integral of both sides to get lny==x+2lnx+C. What I don't understand is how ...
0
votes
1answer
29 views

How many subsets of unknowns whose sum can be determined by the underdetermined system $Ax=b$ with $A \in \{0,1\}^{m \times n}$

Consider a underdetermined system $Ax=b$ with $A \in \{0,1\}^{m \times n}$ (i.e. being a binary matrix), $x \in \mathbb{R}^n$ and $b \in \mathbb{R}^m$. I want find a set $S$, $e \in S$ if and only if ...
2
votes
1answer
34 views

Why does a quaternion rotation matrix simplify to this?

I'm reading Ken Shoemake's explanation of quaternions in David Eberly's book Game Physics. In it, he defines the rotation matrix for a quaternion $q = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} + ...
0
votes
0answers
23 views

Existence of solutions of a $2\times 5$ linear system with two pivot positions

A is a $2\times5$ matrix with two pivot positions. Does the equation Ax=b have at least one solution for every possible b? If the answer is yes, then I understand why. If the answer is no, please ...
1
vote
0answers
24 views

Gauss-Seidel iterative method not converging

I'm trying to solve a linear system of linear equations using the Gauss-Seidel iterative method. I'm writing c code to do it for me since I have over 349 entries to solve. In other words, I have 349 ...
1
vote
1answer
50 views

How to solve a system of equations with sin and cos? [closed]

$50= 35\cos(x) +25\cos(y) $ $0= 35\sin(x)+25\sin(y) $ Thanks!
-3
votes
2answers
54 views

Exponential simultaneous equations, short multiplication formulas - only for sneaky people

I am obligated to do some exercises from some Russian maths book and I solved them, but the teacher told us to use the smartest possible way to achieve this and I guess mine aren't sneaky enough and ...