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.

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2
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1answer
22 views

Finding real $(x,y)$ solutions that satisfies a system of equation.

I was given: $x + y^2 = y^3 ...(i) \\ y + x^2 = x^3...(ii)$ And was asked to find real $(x,y)$ solutions that satisfy the equation. I substracted $(i)$ by $(ii)$: $x^3 - y^3 + y^2 - x^2 + x - y = 0$ ...
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3answers
41 views

Geometric sequence problem including sum of the numbers

Numbers: $a,b,c,d$ generate geometric sequence and $a+b+c+d=-40. $ Find these numbers if $a^2+b^2+c^2+d^2=3280$ I tried this problem and I have system of equations which I can't solve. I think there ...
1
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1answer
42 views

Coefficient matrix in linear equations

I have only started linear algebra, I was watching the lectures of Gilbert Strang, he gave us two linear equation $2x-y=0$ $x-2y=3$ . He wrote this in the way of coefficient matrices and it became $\...
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1answer
22 views

Find the border radius tangent's relative coordinates in a rhombus

I have this rhombus. The width w and height h, and the coordinates of M the intersection of ...
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1answer
34 views

How to solve this ODE's?

I have here this problem, where I want to determine the general solution: $$ u'= - \frac{2v}{t^2}+ te^t $$ $$ v'=-u+t $$ $ t \in \mathbb{R}^+ $ My idea is here to use variation of parameters, but how ...
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2answers
71 views

When do roots of three quadratic polynomials multiply to 1?

Say I have a trio of quadratic polynomials $p_1,p_2,p_3$. Under what conditions will I have $r_1r_2r_3 = 1$ where each $p_i(r_i) = 0$? In other words, when does the following nonlinear system have a ...
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0answers
41 views

Solving 1 equation with 4 variables

In my research, I came up with this equation which consists of 4 variables (t,m,g,h). $$4tm(t-m)(t+m)=h(\sqrt{3}g-h)(\sqrt{3}g+h)$$ Initially, I attempted in equating each product on LHS and RHS, as ...
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2answers
32 views

Finding the maximum value given two system of equations

I was given $(x,y)$ that satisfies both of this equation: $4|xy| - y^2 - 2 = 0\\(2x+y)^2 + 4x^2 = 2$ And was asked to find the maximum value of $4x + y$. Solving for $y^2$, I get this equation: $8x^2 +...
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3answers
69 views

Solve the system of equations for $x$ and $y$?

I'm trying to solve this system of linear equations: $3x^2 - 12y = 0$ $24y^2 -12x = 0$ for $x$ and $y$, but I'm a little confused. I get $x = 0, 2$ and when I plug those into my first equation I get $...
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3answers
47 views

Finding “hidden” solutions to a simple complex system of linear equations

I have been trying to solve the following system of linear equations in the complex plane: $$\begin{cases} z_1 = -iz_2 \\ z_2 = iz_1 \end{cases} $$ I know the solution, it's $z_1 = 1, \space z_2 = i$, ...
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0answers
15 views

How to decouple these system of 2nd order partial differential equations?

Solving a problem I found this system of equations: $$ (\partial _t ^2 + \partial _x ^2 + \partial_y ^2 + \partial_z ^2)a_1(\vec{x},t) - 4gB(x\partial_y - y\partial_x)a_{\color{Red}2}(\vec{x},t) - 4g^...
1
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1answer
35 views

Is there a word for a contradictory set of linear system of equations?

So in basic math, we tend to learn that we can solve for the variables if there are n equations and n unknowns. But let's say the equations are contradictory, for example, $x + y = 1$ and $x + y = 5$....
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0answers
58 views

Writing Math equations in MSE. [migrated]

How do I write the mathematical equations in a question, here on stack exchange? When I try to write a question which involves equations, I get stuck as I don't know how to input those equations... ...
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1answer
43 views

Why the system of equation has a nontrivial solution?

I am studying barycentric coordinate, and I encountered the following from my book: Consider two parallel lines $u_1x + v_1y + w_1z = 0$ and $u_2x + v_2y + w_2z = 0$. Because they are parallel, we ...
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2answers
13 views

When to ignore the 0 values of a variable when dividing an equation by a variable

So, I was watching this video of parametric equations and the elimination of the parameter. When you are solving a system of equations,you basically cannot divide by any of the variables, in case that ...
3
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3answers
81 views

If $a+b+c=7$ and $\frac1{a+b}+\frac1{b+c}+\frac1{c+a}=\frac7{10}$, then evaluate $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}$. [duplicate]

Let $a$, $b$ and $c$ $\in \mathbb{Q}$ such that $a+b+c=7$ and $\cfrac1{a+b}+\cfrac1{b+c}+\cfrac1{c+a}=\cfrac7{10}$ What does $\cfrac a{b+c}+\cfrac b{c+a}+\cfrac c{a+b}$ equal to? The final result is $...
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1answer
23 views

Solve the following non-linear system of equations $x = \alpha \log(y/(1-y)), y = \alpha \log(x/(1-x)) - \beta$ in terms of $\alpha, \beta$

I have the following non-linear equations, $$x = \alpha \log(y/(1-y))$$ $$y = \alpha \log(x/(1-x)) - \beta $$ where $\alpha, \beta$ are constants, $\log$ is the natural logarithm. I wish to solve for $...
2
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1answer
59 views

Solving system of modulo inequalities?

I've already asked similar question, but back then I didn't know about modulo arithmetic. I think I can ask it more clearly now. Is this type of system of equations solvable? If yes, how? $$ \begin{...
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0answers
49 views
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0answers
18 views

Why does requresive least squares need to have a variance onto the input signal?

Requresive Least Squares(RLS) is a famous algorithm for solving linear system and it's used a lot with system identification for finding parameters for a polynomial function: $$yA(q) = uB(q) + eC(q)$$ ...
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0answers
13 views

determine the orthogonal matrices with the variation of two parameters

I have the following exercise: Determine the orthogonal matrices of the type $$ \begin{matrix} a+1 & 0\\ 0 & b\\ \end{matrix} $$ with $a,b \in\mathbb R $ I thought to proceed ...
2
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1answer
47 views

Are the only solutions to this implicit functional equation linear?

Let $f:(0,1] \to (-\infty,0]$ be a smooth function which is strictly negative on $(0,1)$ and satisfies $f(1)=0$. Let $\epsilon \in (0,1)$ and let $x,y:(0,\epsilon) \to (0,1]$ be smooth functions ...
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0answers
44 views

Solving Linear System of Equation in Polynomial Time

Let $ \mathbf{b} \in \mathbb{R}_+^n$, $\mathbf{E} \in \mathbb{R}_+^{n \times m}$, $\mathbf{V} \in \mathbb{R}_+^{n \times m}$ with $\mathbf{E}^T \mathbf{1}_n = \mathbf{1}_m$ and $\mathbf{V} \mathbf{1}...
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0answers
15 views

How to solve these two equations together to get an expression for $V_{OUT}$ and $I_{DS}$ [closed]

$$I_{DS} = \frac{K}{2} (V_{IN} - V_{OUT} - V_T)^2$$ $$V_{OUT} = I_{DS} R_S$$ I have these two equations which I want to solve simultaneously to get an expression for $I_{DS}$ in terms of $V_{IN}$, $K$...
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0answers
38 views

Solution to this system of ODEs

I'm not an expert in ODEs so this might be a dumb or a non-rigorous question. I have a system of matrix-vector ODEs that I want to solve, which is of the form: $$ \dot f(t) = G(t)v(t),\quad \dot G(t) =...
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0answers
24 views

Finding the initial conditions for system of ODE

How to find out the the initial conditions for the below system of first order autonomous differential equation i.e $x_1(0)=? , x_2(0)=?, x_3(0)=? \\ \frac{dx_1}{dn} = -4-3x_1+2x_3-x_1x_3+x_1^2 \\ \...
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0answers
49 views

How to solve the following system of linear differential equation with variable coefficients.

I have tried with some methods but cannot proceed the following system of differential equations: $$\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}v_1(t)=-v_1(t) \cos(t)\sin(t)+v_2(t) \sin^2(t)\\ \...
1
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1answer
59 views

Closed Form Solution to System of Equations

Let $ \mathbf{b} \in \mathbb{R}_+^n$, $\mathbf{V} \in \mathbb{R}_+^{n \times m}$ with $\mathbf{V} \mathbf{1}_m = \mathbf{1}_n$ where $\mathbf{1}_n$ is the vector of ones of size $n$. I have the ...
0
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1answer
39 views

How to compute values of variables of a system of equations? [closed]

I have some equations which relate to each other and these are down below $z+p=12.78248$ $a+x=2.16943$ $z+2a=11.75629$ $p+2x=5.36505$ I am trying to figure out a numerical value of any one variable ...
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1answer
38 views

Solving system of equations with three unknowns

I need to solve an equation of a line using three known coordinate pairs (x0, y0), (x1, y1), and (x2, y2). The equation of the plane is, of course, ax + by + c = 0. I'm writing a little piece of code ...
2
votes
1answer
35 views

Trigonometric system for $(0,\pi/2)$

If $x,y,z\in (0,\frac{\pi}2 )$, find all solutions to: $$\begin{cases} \tan x+\sin y+\sin z=3x \\ \sin x+\tan y+\sin z=3y \\ \sin x+\sin y+\tan z=3z\end{cases}$$ Can someone give a hint to me? I can't ...
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0answers
25 views

Solving a system of equations, based on a gear, such that solutions are natural numbers or proving no solution exists.

I'm not sure how to exactly ask this question. I'm essentially looking for a way to solve a set of natural numbers from a set of equation based on a gear. In addition I'm looking for a way to prove if ...
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0answers
24 views

For the parametric curve whose equation is $x = 4\cos θ$ and $Y = 4\sin θ$, what would be the concavity of curve at θ = π/4?

What I'm actually confused is that I found the slope to be -1 which is correct but when for finding the concavity I take the double derivative of the given function it gives the answer tan π/4 which ...
-1
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3answers
52 views

solve equations having term $ xy$ [closed]

I want to solve equations: $$x^3-3xy^2=-11$$ and $$y^3-3x^2y=-2$$ for $x$ and $y$. How can I do it?
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1answer
40 views

A tricky nonlinear pair of equations [closed]

For example, a question like this. Solve $$ \begin{cases} \frac1x+\frac1{2y}&=(x^2 + 3y^2)(3x^2 + y^2) \\ \frac1x-\frac1{2y}&=2(y^4 - x^4) \\ \end{cases} $$ I couldn't see any way to approach ...
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2answers
48 views

solve $x_1'(t) =x_1 (t) + 2x_2(t) , x_2'(t)=2x_1(t)+4x_2(t)$

I have found eigenvalues and eigenvectors, but I have no idea what to do next. How do I find x with them?
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1answer
24 views

Solving Modular System with 2 different Moduli

Is there any way to solve for $a$ and $b$ in: $$ a*b \equiv s_0 \mod r_0 $$ $$ a - b - a*b \equiv s_1 \mod ( r_0 - 1) $$ I have the values of $r_0$, $s_0$, $s_1$ and would ...
1
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1answer
16 views

System of equations of non-relativistic scattering in the laboratory system

Considering the system of equations of non-relativistic scattering in the laboratory system: $$\begin{cases} \dfrac{1}{2} m_{1} v_{1}^{2} &=\dfrac{1}{2} m_{1} v_{2}^{2}+T_2 \,,\\ m_{1} v_{1} &=...
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0answers
23 views

Solving Linear Modular System with different Moduli

Suppose we had a list of modular expressions as follows: $$ y_0 \equiv s_0 \mod r_0 $$ $$ y_0 \equiv s_1 \mod r_0-1 $$ $$ y_0 \equiv s_2 \mod r_0-2 $$ $$ y_0 \equiv s_3 \mod r_0-3 $$ $$ y_0 \equiv s_n ...
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0answers
34 views

Finding number of solutions?

The number of solutions of the equations $x_{2}-x_{3}=1$ $-x_{1}+2x_{3}=2$ $x_{1}-2 x_{2}=3$ (a) zero (b) one (c) two (d) infinite In a solution,I saw $$ \begin{array}{l} \text { Let } \Delta=\left|\...
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0answers
22 views

Identity coming from system of polynomial equations

I found a strange identity in a system of polynomial equations I am studying, and I do not know how to verify it. The system has $n+3\binom{n}{2}$ unknowns, represented by the variables $(\{x_j\}_{j=...
0
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2answers
57 views

System of equations with infinitely many solutions

Consider this system of equations $$2x + 3y + z = 6$$ $$-x+y+2z = 7$$ $$ax+y+4z=b$$ Find the values of $a$ and $b$ for which the system has an infinite number of solutions. I am stuck struggling with ...
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1answer
13 views

Question based on Treating System of Linear equations at Linear Functional

While self studying Linear Algebra from Hoffman and Kunze, I have following question. I have a question in blue highlighted line ( Page 103 of book) How does in the underlined line, n- tuple gives ...
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0answers
42 views

What objective do gradient descent and steepest descent maximize when solving linear systems with infinite solutions?

When we use various local heuristic optimization methods for the solving of linear systems $Ax=b$ (e.g. gradient descent, steepest descent ($L_1$ norm), conjugate gradients), we implicitly are trying ...
1
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3answers
68 views

System of equations and recurrence relation

I am trying to find the general solution for $N$ of the following system of equations $$ \begin{cases} (x_n - x_{n-1})^2 + (y_n - y_{n-1})^2 = \left(\frac{\theta}{N}\right)^2 \\ {x_n}^2 + {y_n}...
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2answers
30 views

Proving Linearly Independence from System of Equation

I am trying to understand the proof of Linearly Independence of the basis set $\{1, x, x^2, x^3\}$. It is written that - Substituting $3$ other values of $x$ into the above equation yields a system ...
1
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1answer
46 views

Need help in solving a linear algebra ( System of Equations) quiz problem

I am solving previous year quiz problem of my class and I am unable to solve this question in linear algebra. Let $A \in M_{m \times n}(\Bbb{R})$ and let $b_0 \in \Bbb{R}^m$. Suppose the system of ...
1
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1answer
40 views

Deriving formula for cross-product.

It is given on pg. #106, 107 in the book by: Thomas Banchoff, John Wermer; titled: Linear Algebra Through Geometry, second edn.. Consider a system of two equations in three unknowns: $$a_1x_1 + a_2x_2 ...
2
votes
2answers
133 views

Advanced Ratio Problem

I have a video game(Factorio, if you have heard of it) that I like to play that involves many different kinds of ratios in order to optimize the production of items that require other items. Normally ...

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