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.
8,555 questions
-1
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0
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76
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A $4\times 4$ linear system with four parameters
Consider the following linear system
$$\begin{cases}
x + y + z + t = 1,\\
ax + by + cz + dt = 1,\\
a^2x + b^2y + c^2z + d^2t = 1,\\
a^3x + b^3y + c^3z + d^3t = 1\end{cases}$$
What is ...
1
vote
0
answers
24
views
Singular integral equations for beginners
I think I just got past the initial peak of the Dunning-Kruger plot with Singular Integral Equations applied to fracture mechanics (i.e., I realized I am lacking a lot of knowledge) and I thus would ...
-3
votes
0
answers
44
views
Geometry Problem About Tangent [closed]
Let C1:(x-y)²-8(x+y-2)=0 and C2:4x²+9y²=36 be two curves on x-y plane. If A,B be two common points on C1 from where perpendicular tangents are drawn to C2.
Then, if coordinates of A(a1,b1) and B(a2,b2)...
- 1
1
vote
0
answers
41
views
Finding $x$ at intersection of $a \equiv bx\mod M$ and $c \equiv bx\mod N$ [duplicate]
I have two equations $a \equiv bx\mod M$ and $c \equiv bx\mod N$ in which I know $a, b, c, M$, and $N$. I am trying to find $MN > x \geq 0$ ($x$ is non-negative). Further, $M$ and $N$ are odd and ...
- 685
-1
votes
1
answer
62
views
Finding Pythagorean triples divisible into smaller Pythagorean Triples.
In this question, I was able to show how some Pythagorean triples can be divided into two smaller Pythagorean triples by the perpendicular from the hypotenuse to the right angle... as shown in the ...
- 6,614
1
vote
0
answers
51
views
Why does implicit differentiation fail when using it to solve an equation?
Suppose that I have two real-valued functions
$f(x):=\cos(\frac{\pi}{2}x)$ and $g(x):=x^2 - 1$
and I want to find their intersection points when both are graphed on the Cartesian plane.
For each ...
- 83
-1
votes
0
answers
21
views
How to obtain an isolated term from an expression [closed]
I have the following equation and I want to rearrange it to obtain isolated $L$ (a closed form for $L$). How can I do it?
$$
\sum_{i \in J} \frac{L c_i v}{2t^3} = \sum_{i \in J} c_iv\frac{\sqrt{4t^2 + ...
- 1
1
vote
0
answers
56
views
Why is $X e^{\Lambda t}X^{-1}$ the general solution to system of $n$ linear first order ordinary differential equations?
I know that a system of n first order linear ordinary differential equations:
$$
\begin{align}
\begin{bmatrix} \frac{dy_1}{dt} \\ \vdots \\ \frac{d^ny_n}{dt^n} \end{bmatrix} &= \begin{bmatrix} a _{...
- 33
-2
votes
0
answers
33
views
Find equation(s) that satisfies all requirements [closed]
Which equation(s) for y(x) satisify these requirements?
y(0)=0
y(x1)=y2
y(x_max)=y_max
y'(x_max)=0
y'(x)>0 for 0<x<x_max
y''(x)<0 for 0<x<x_max
Example numbers:
(x1, y2) = (2, 7)
(...
- 1
0
votes
0
answers
83
views
System of two equations in two variables involving $\sin$ and $\cos$ [closed]
I’m trying to find a symbolic solution to this system of two equations in two variables, $\theta_1$ and $\theta_2$, without any success:
$$
\left\{\begin{array}{rcl}
{\displaystyle l_{1}\sin\left(\...
0
votes
1
answer
77
views
How to find a condition on $a,b,c$ such that solutions exist to a system of inequalities
How can I find a condition on $a,b,c \in \mathbb{R_{> 0}}$ such that there exist solutions $x,y,z \in \mathbb{R}$
$$
\begin{align}
ax-y-z&>0 \\
-x+by-z&>0 \\
-x-y+cz&>0
\end{...
2
votes
1
answer
66
views
Algebraic Equation with multiple variables.
A lot of times I have multiple equations, and I don't have to completely solve them, rather just transform them into a particular form and get its value. For that I need algebraic manipulations and ...
7
votes
2
answers
732
views
Strange ODE system
As I was solving a physics problem I came across this very strange ODE. My goal is to get the differential equation for the temperature T. The problem is that I can't get rid of the current I in any ...
- 151
0
votes
1
answer
32
views
Proving a pattern for a system of equations of size N
I've got a system of equations of size N. Here's its definition:
Definition
Let N be an integer $\geq 2$. For N, this system has N equations and N variables. I'll define the variables as $a_0, a_1, ...
- 125
0
votes
0
answers
27
views
Error bounds for ill-conditioned linear systems
I have two ill-conditioned linear systems,
$$
P_X = P_Yq \\
\hat{P}_X = \hat{P}_Y \hat{q},
$$
where the left-hand sides are column vectors, $P_Y$ and $\hat{P}_Y$ are non-square matrices and where I ...
- 5
0
votes
0
answers
67
views
A question on system of linear equations.
Let $A$ be a $3×4$ matrix and $b$ be a $3×1$
matrix with
integer entries. Suppose that the system
Ax=b has a complex solution. Then which statements are true?
Ax=b has an integer solution
Ax=b has a ...
- 11
2
votes
3
answers
72
views
Trouble solving two simultaneous equations, $ab = \frac{1}{2}$ and $a^2 - b^2 = 0$
I am trying to solve the following simultaeneous equations:
$$ab = \frac{1}{2}$$
$$a^2 - b^2 = 0$$
Expanding #2 gives us $(a+b)(a-b) = 0$, which in turn yields $a+b=0$ or $a-b=0$. Thus:
$$a=-b\text{ ...
- 31
1
vote
1
answer
81
views
General solution to this coupled system of differential equations
I am trying to solve a circuit for the voltages at specific nodes as a function of time. I've made some changes to the problem given in as a textbook problem.
I have the following system of ...
- 11
0
votes
2
answers
89
views
Solving the system $xy = 15$, $x + y - \sqrt {\frac{{x + y}}{{x - y}}} = \frac{{12}}{{x - y}}$
The system is:
$$\begin{align}
x + y - \sqrt {\frac{{x + y}}{{x - y}}} &= \frac{{12}}{{x - y}}\\[4pt]
xy &= 15
\end{align}$$
I've tried making the following substitutions: $t = x + y,u = x - ...
1
vote
1
answer
44
views
Solve this linear system without explicitly inverting the matrix
I have a general system of linear equations which looks as follows:
$$\left(1-\frac{\hat{A}}{8}-\frac{3\hat{A}}{8}\left(1+\frac{\hat{A}}{6}\right)^{-1}\left(1-\frac{\hat{A}}{6}\right)\right)X=B$$
...
- 32.2k
-1
votes
1
answer
49
views
Finding all positive reals $r$ satisfying $y_1=\frac{N}{2(1+r)}$ and $y_2=\frac{N}{2(1+1/r)}$ for positive integers $y_1$, $y_2$, $N$
$y_1$, $y_2$ and $N$ are all positive integers. Given you know the value of $N$, find all the positive values of $r$ ($ r\in \mathbb{R^+})$ that satisfy the following two equations.
\begin{equation}
...
- 133
2
votes
1
answer
60
views
Solving a recurrence relation using roots of a quadratic equation
if the roots of the quadratic equation $x^2 = px + q$ are $\alpha,\beta$. since they are roots,
$$\alpha^2=p \alpha+q$$
$$\alpha^n=p \alpha^{n-1}+q \alpha^{n-2}$$
similarly can be proven from $\beta$, ...
- 121
0
votes
2
answers
63
views
Stack Double problem where answer should be only one
Question:
$7$ men labeled $A−G$ play a coin flip game that starts with player $A$. At each turn, the player will flip a coin. If it appears heads, then the player must double the stack of each other ...
- 5,489
2
votes
1
answer
96
views
How to combine two equations?
I've participated in the German Math Olympiad a few days ago (this stage is alread over) and there's the following problem:
$$x^2 = 21 - 4 \cdot y$$
$$y^2 = 21 + 4 \cdot x$$
where all real solution to ...
- 21
4
votes
3
answers
380
views
Is there an elegant general method for solving linear multiplicative system of equations in modulo 2? Here is an interesting example problem.
Here is the following problem:
I have solved the system of equations with simply using brute force but I feel there must be a ...
- 41
0
votes
0
answers
16
views
Parameterization of Kasner exponents
The Kasner metric is given by,
\begin{equation} \tag{1} \label{1}
ds^2 = -dt^2 + t^{2 p_1} dx^2 + t^{2 p_2} dy^2 + t^{2 p_3} dz^2
\end{equation}
where $p_1, p_2, p_3$ are the Kasner exponents. ...
- 523
0
votes
0
answers
21
views
Complex Step coefficients
Asking a very similar question to this one Finite differences coefficients. I was wondering if there existed a similar method for obtaining the coefficients of the complex step approximation. You can ...
- 1
0
votes
1
answer
37
views
About the number of solutions of system of non linear equations
I was solving a problem of Lagrange multipliers and faced the following system of non-linear equations:
I managed to find 6 solutions by setting each of the variable to 0 and then playing arround ...
- 143
-2
votes
0
answers
49
views
A system that I didn't solve
We have that $s_1=a+bi$, $s_2=a-bi$, $L>0$ and $$
s_3\in\mathbb{R},\qquad s_4=s_3+\frac{2\pi}{L}k,\qquad s_5=s_3+\frac{2\pi}{L}(k+l),\qquad k,l\in\mathbb{N}.
$$
I want to know what is $L, s_3, a,b$ ...
- 135
-1
votes
1
answer
81
views
Solving polynomial system of equations [closed]
Context: I have to solve the following system of three equations that are in descending powers of $\alpha$:
$$\begin{cases}0&=&A\alpha^2+B\alpha+C\\ 0&=&\frac13 A\alpha^3+\frac12 B\...
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0
votes
1
answer
40
views
Alternative solution: if $f(x, y) = (mx+y, 2x+my)$ is surjective, what is $m$?
$\begin{align} f: \mathbb{R^2} &\to \mathbb{R^2} \\ (x,y) &\mapsto (mx +y, 2x + my) \end{align}$
$f$ is surjective. Find $m$.
Hi, I've just begun my study in linear algebra in college. We've ...
- 1,642
1
vote
0
answers
24
views
Motion word problem
Two cyclists depart from two different locations A and B at the same time. They cycle towards each other at constant speed (km/min) and meet after time t. If the first cyclist's speed was twice as big,...
3
votes
2
answers
52
views
Rewriting the system $x'=\sqrt{|y|}+x^2+h(t)$, $\;y'=e^x+y$ in an equation of second order
I have to rewrite the system of
$$\begin{align}
x' &= \sqrt{\left|y \right|} + x^{2} + h(t)\\
y' &= e^{x} + y
\end{align}$$
in an equation of second order.
I know that I have to find $ x'' $ ...
- 301
0
votes
0
answers
25
views
How to solve this set of simultaneous equations to smoothly "weld" 2 functions together?
I have a function $\Phi(\rho)$ that is approximated well if I split it into 2 regimes, one for small values of $\rho$, which I will call $\Phi_S$ and one for large values of $\rho$, which I will call $...
- 23
0
votes
2
answers
83
views
Three equations three unknowns [closed]
I have the following set of equations with a,b,c being real numbers
$$a^2 (b+c)= 2$$
$$b^2 (c+a)=4$$
$$c^2 (a+b)= 10$$
The question wants me to solve for all the possible values of the product abc.
...
- 399
0
votes
1
answer
31
views
System of difference equations with complex numbers
Solve the equation $z_{n} = \overline{z_{n}} + iz_{n}$ for $z_{n} \in \mathbb{C}$.
I´ve gotten this far. Put $z_{n} = x_{n} + iy_{n}$ where $x_{n}, y_{n} \in \mathbb{R}$. Then
$$
x_{n+1} + y_{n+1} = ...
- 305
0
votes
0
answers
90
views
How can I solve $y+x^2=0$ and $\frac{x^x}{2-\sqrt{y}}=\frac{1}{\sqrt{e^{\pi }}}$
A few months ago, I wrote down this set of simultaneous equations, hoping there might be a way to solve them:
$y+x^2=0$
$\frac{x^x}{2-\sqrt{y}}=\frac{1}{\sqrt{e^{\pi }}}$
I know that one solution to ...
- 351
-1
votes
1
answer
46
views
Finding a Polynomial Equation in $x$ and $y$. Given $x=a+b$ and $y=ab$.
I am working with the relationships $x=a+b$ and $y=ab$, where $a,b,x$ and $y$ are real variables. I would like to know if it is possible to derive any polynomial equation $p(x,y)=0$ that relates $x$ ...
- 71
5
votes
3
answers
266
views
Approximate the first solution of $\sum_{k=0}^n \frac 1{x-k}=0$
I would like to obtain formally a "good" approximation of the first root of function
$$F_n(x)=\sum_{k=0}^n \frac 1{x-k} $$ where $n$ can be extremely large.
For this problem, as earlier when ...
- 274k
4
votes
1
answer
83
views
About a functional equation
Problem: Let $(f, g)$ be a pair of function that satisfies the following:
$f, g: \mathbb{R}^+ \rightarrow \mathbb{R}^+$
$\forall x, y \in \mathbb{R}^+, f(xy + f(x)) = xf(y) + g(x)$
Find all possible ...
- 141
0
votes
0
answers
37
views
Number of $m\times n$ matrices of rank $0\leq k\leq min\{m,n\}$
I am studying the enumeration of $m$-by-$n$ matrices over a finite field $F_q$ with a specific rank $r$. I came across a post that describes the number of such matrices using the formulas given in the ...
- 173
1
vote
1
answer
158
views
Solving the system $x-x^2=az^2$, $\;y-y^2=2xy+bz^2$, $\;1-z=2x+2\alpha y$ for $x$, $y$, $z$
I have the following system I need to solve:
$$\begin{align}
x-x^2 &=az^2 \\
y-y^2 &=2xy+bz^2 \\
1-z\phantom{^2} &=2x+2\alpha y
\end{align}$$
where $a$, $b$, $\alpha$ are just constants.
...
- 1,880
1
vote
0
answers
30
views
Probability of a Linearly Independent Subset in Random GF(2) Equations
Assume that, over $GF(2)$, an algorithm takes $k$ variables as input and generates $6$ linear equations. Each of these $6$ linear equations consists of up to $2^k-1$ terms (excluding the constant term)...
- 173
0
votes
1
answer
31
views
Random Binary Matrix Column Space with XOR Addition in Regards to Random Binary Vector
Given that you have a random binary matrix $A$, which is has dimensions $(n,l)$ where $n>l$ (has more rows than columns), is it possible that a random binary vector $\vec{b}$ is in the column space ...
- 101
-3
votes
1
answer
55
views
How to represent x and y in terms of a and b? [closed]
For given, x an y and we get equations of a and b below:
m = 3*(x^2)/2*y
a = m^2 - 2*x
b = m*(x-a) - y
How to do it reverse? How to represent x and y in terms of ...
- 127
0
votes
1
answer
77
views
How to calculate $p_{A}+ p_{B}$ quickly?
Given $p_{A}, p_{B}, p_{C}$ so that $p_{A}+ p_{B}+ p_{C}= 1$, and the transition matrix equation $\begin{bmatrix} a+ m & -b & -s\\ -a & b+ d & 0\\ -m & -d & s\\ \end{bmatrix}\...
- 100
0
votes
0
answers
9
views
Efficient solution of many systems of equations involving Hadamard powers
I have many matrix equations of the form
$$A^{\circ z_i}\mathbf{x}_i=\mathbf{b}_i$$
where ${\circ z_i}$ indicates the Hadamard power (element-wise). $A$ is a very large dense symmetric matrix with the ...
- 341
4
votes
3
answers
263
views
How to solve the following systems of equations: $3x^{2}+y=7$ and $4y^{2}-x=5$
How to solve the system of equations :
$3x^{2}+y=7$ ; $4y^{2}-x=5$
From desmos it is very much clear to me that the given system of equations will have $4$ solutions.
Now
I am trying to solve the ...
- 67
1
vote
1
answer
104
views
Projective lines and minors of matrices
Consider the $3\times (n+1)$ matrix
\begin{align*}
\begin{pmatrix}
x_0 & x_1 & ... & x_n\\
a_0 & a_1 & ... & a_n\\
b_0 & b_1 & ... & b_n
\end{pmatrix},
\end{align*}
...
- 1,173
1
vote
1
answer
90
views
Solving systems of equations over arbitrary algebraically closed field
When considering a system of equations over an arbitrary algebraically closed field, does the number of solutions depend on the field? (As long as the equations do not "degenerate"?)
(The ...
- 1,173