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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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2answers
13 views

How can I transform this expression into the desired one?

I know that (x/119)*19 is equal to x - (x/1.19), for every value of x. However, I am trying to figure out, for curiosity, what steps do I take from the expression (x/119)*19 to reach the expression x ...
0
votes
1answer
23 views

System of equations having unique solutions [on hold]

Question is, for what value of K will the system of equations $x+y+z=2$ $2x+y-z=3$ $3x+2y+kz=4.$ have a unique solution.
2
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0answers
17 views

Sustainable population - When to breed

I would like advice on an approach to solving this real world problem. I'm not certain if the solution is to solve a system of equations, or something else. The problem is stated below with the ...
0
votes
0answers
21 views

Explicit solution of $XA=BX$

Consider the equation $$XA=BX,$$ where $X,A,B$ could be matrices or (pseudo) differential operators. How to represent $X$ using $A,B$, that is $$X=f(A,B),$$ where $f$ is a known function.
1
vote
1answer
32 views

A system of PDEs

Consider the following system for $u(x,y)$, $v(x,y)$. $$2x^2yu_x + 5xy^2u_y + 2x^2y^2v_y + 5xyu + x = yu_y - x^2v_x + u - 2xv = 0 $$ Prove that it is equivalent to a second order semilinear PDE. ...
0
votes
2answers
23 views

How do I interpret the results of Gaussian elimination of two linear systems that have the same coefficient matrix?

The systems of linear equations $$x_1 - x_2 + 2x_3 - x_4= 6$$ $$x_1 - x_3 + x_4 = 4$$ $$2x_1 + x_2 + 3x_3 - 4x_4 = -2$$ $$-x_2 + x_3 - x_4 = 5$$ and $$x_1 - x_2 + 2x_3 - x_4= 1$$ $$x_1 - x_3 + x_4 = ...
0
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0answers
31 views

Condition on existence of solution for system of inequalities

Let $a,b,c,x,y,z>0$. Under which condition of $a,b,c$ there will exists some $x,y,z>0$ such that \begin{equation} \begin{cases} x - \dfrac{a}{2} - c \left( 4x^2 + y^2 + z^2 \right) & >...
0
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4answers
80 views

4 simultaneous equations in real numbers [on hold]

Solve the following system of equations in real numbers: $\begin{cases}x^2+zx=y+z\\y^2+xy=z+x\\z^2+yz=x+y\\xyz=1\end{cases}$
-3
votes
3answers
37 views

ABC triangle has vertices $A(3,5) B(-7,1)$ and $C(5,5)$. Write the equation of the median from vertex A. [on hold]

The answer is $A$: $x-2y+7$ according to the book, but I want to know how to solve it. Please show the solution step-by-step.
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votes
1answer
28 views

Solving a system of equations for $x, y,$ and $T$

While solving a computer science question, I got a system of equations$$\begin{cases}x^2+(100-y)^2=\frac TA\\(100+x)^2+(100+y)^2=\frac TB\\(100-x)^2+(100+y)^2=\frac TC\end{cases}$$ However, I'm ...
0
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0answers
21 views

the set of all solutions of the system $ax+by+cz =0, dx+ey+cz=0$ forms a subspace.

To show that the set of all solutions of the system $ax+by+cz =0, dx+ey+cz=0$ forms a subspace. Subtracting we get $x = \frac{e-b}{a-d}y$ and putting this in any one of the equations we get $z= \...
0
votes
0answers
33 views

Determining boundaries separating modes of behavior in a system of differential equations

By way of example, suppose I have a phase diagram illustrating the following system of differential equations... $dx/dt = (1-x^2-2y)x$ $dy/dt = (1-2x-y)y$ ...as follows: The phase diagram is drawn ...
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0answers
13 views

Solve Least Squares with Constraints

I have a problem where I am supposed to solve a system of equations in matrix form. The system is 4x4 with 4 unknows. The matrix comes from the least squares method. However, I have a constraint. It ...
1
vote
1answer
45 views

Find exhaustive range of $k$ such that $f(x)=\frac{x-1}{k-x^2}$ never belongs to $\left[-1 \:\: \frac{-1}{3}\right]$

Find exhaustive range of $k$ such that $$f(x)=\frac{x-1}{k-x^2}$$ never belongs to $\left[-1 \:\: \frac{-1}{3}\right]$ My try: Letting $$y=\frac{x-1}{k-x^2}$$ we get $$yx^2+x-(1+ky)=0$$ and since $...
1
vote
0answers
35 views

How can apply a perturbation method in a system of equations?

If a have a system of equation $$ \frac{d u}{dt} = 1- u e^{\epsilon(q-1)}$$ $$ \frac{d q}{dt} = u e^{\epsilon(q-1)}-q$$ $u(0)=q(0)= 0$ How Can I apply the perturbation methot $\epsilon$ small ...
1
vote
0answers
29 views

Solving 2 simultaneous equations with indeces

I have 3 formulae used to calculate some probabilities of some events, 0, 1, 2 occurring. P($X=0$) = $(1-$z$)^R$ $\cdot$ $z^A$ P(X = 1) = $0.5^{R+A}$ P(X = 2) = $z^R$ $\cdot$ $(1-$z$)^A$ Where $...
0
votes
0answers
16 views

given a system of N linear equations, Is there an algorithm that can find a solution that solves the most number of equations in this system

My apologies if this question makes no sense; I am trying to find an algorithm that can solve a linear system of equations. Unlike most problems like this- for this particular case, this algorithm ...
0
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0answers
12 views

Converting a linear difference model to matrix form and using matlab to predict next 20 generations

I've been looking at ODEs and population modeling, I have a problem with the linear difference model of populations $A_n$ and $B_n$ $A_{n+1}=(0.99+(0.25)(0.01))A_n +(0.25)(0.05)B_n = 0.9925A_n + 0....
0
votes
1answer
59 views

Find roots of the equation $y=2 \sin(3x+40), \;x \in (-2\pi, 2\pi)$

Find the roots of the equation: $$y=2 \sin(3x+40), \;\;x \in (-2\pi, 2\pi)$$ In the book given that there are 12 roots exist. I am able get only 2 roots. Could anyone explain?
1
vote
3answers
35 views

Find the cubic polynomial f(x) such that

Find the cubic polynomial f(x) such that $𝑓(−1)=-4$, $𝑓'(−1)=4$, $𝑓''(−1)=-6$, $𝑓'''(−1)=12$ I know that i have to use an augmented matrix in order to solve it, so i decided to use the ...
0
votes
3answers
56 views
0
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0answers
17 views

Solving a special system of linear algebraic equations

Consider the following function: $$\Omega \left( \mathbf{F},\mathbf{E} \right)={{\left( \text{diag}\left( \mathbf{F} \right)\mathbf{M\sigma }-\mathbf{XE} \right)}^{\text{T}}}\mathbf{W}\left( \text{...
1
vote
1answer
37 views

How to solve a two variable equation

Suppose $m,n$ are positive constants,there is an equation $m|\lambda_1|^2+|\lambda_2|^2-n\lambda_1\bar{\lambda_2}-n\lambda_2\bar{\lambda_1}=0$,where $\lambda_1,\lambda_2$ are nonzero complex numbers. ...
0
votes
2answers
27 views

Find $a,b,c$ if $f(4)=1, f(-2)=14, f(5)=-2$ and $f(x)=ax^2 + bx + c$

Would anyone please explain to me the steps needed in order to find $a,b,c$? I understand I must use elimination, but the answers I keep on getting are very unrealistic. Thanks in advance.
0
votes
0answers
20 views

Possible solutions of some system of equations

What are the possible solutions of $u_1v_1 + u_2 v_2 + u_3 v_3=\min_i u_i$ $v_1 +v_2 + v_3 = 1$ $u_1 +u_2 + u_3 = 1$ $u_i > 0, v_i \geq 0, i=1,2,3$ Two solutions are immediate: 1) $v_{\...
2
votes
1answer
39 views

System of six equations in real numbers

Let $a,b,c,d,e,f$ be real numbers. Solve the following system of equations: $\begin{cases}a+b=-e\\ ab=f\\ c+d=-a\\ cd=b\\ e+f=-c\\ ef=d\end{cases}$ I got stuck trying to solve this problem by ...
0
votes
1answer
19 views

Solving a system of three equations with four unknowns with matrices.

I am new into Linear Algebra and I am trying to solve the following system of equation: $\begin{cases} \begin{matrix} x+2y-z-t=0 \\ \frac { 1 }{ 4 } x-y-\frac { 1 }{ 2 } z+2t=1 \\ 2x-2y-3z+7t=4 \end{...
0
votes
0answers
35 views

Linear algebra rref uniqueness proof exaplanation [closed]

previously proved that $d_1 = d'_1$. Second Step Suppose that we have determined $d_1 = d'_1, d_2 = d'_2 \ ...\ d_p = d'_p.$ Let us now show that $d_{p+1} = d'_{p+1}$ rref uniqueness proof in the ...
0
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2answers
19 views

Parameter Solving System

What method is used here to find the solutions? I've tried to solve it with matrix coefficients but no dice. Is there an immediate way to get to the solutions that I'm missing?
-1
votes
0answers
28 views

How to figure out a quadratic equation if you don't know the x-coordinate

Suppose you somehow know the value of a function y(x) at three different points and you know the slope at that three points. But you don't know the x-coordinate. Is there anyway to figure out a ...
0
votes
0answers
18 views

Calculating edge lengths of a tetrahedron based on constraints

Given a tetrahedron $\Delta$ with unspecified vertices $v_1, v_2, v_3, v_4$, but numerically specified edge lengths $h_{12}, h_{23}, h_{31}, h_{14}, h_{24}, h_{34}$, which create the constraints $h_{...
2
votes
2answers
54 views

Find $A$ if solution to $Ax=b$ is given

Solution to $Ax=b$ is $x=\begin{bmatrix} 1\\ 0\\ 1\\ 0 \end{bmatrix}+\alpha_{1}\begin{bmatrix} 1\\ 1\\ -1\\ 0 \end{bmatrix}+\alpha_{2}\begin{bmatrix} 1\\ 0\\ 1\\ 1\end{bmatrix}$. For $b=(1,2,...
0
votes
1answer
37 views

Solving Systems of Equations (Linear Algebra)

How would I solve this system of equations? $$x_1 - x_2 + x_3 - x_4 = 0 \\ x_1 + 2x_2 + 3x_3 + 4x_4 = 0$$ So far I have made it to: $$x_2 = \frac{-2 x_3}{3} - \frac{5 x_4}{3} \\ x_1 = \frac{5 x_3}{...
1
vote
0answers
18 views

Proving the existence of real solutions to a system of multivariate polynomials for a range of parameters

I need to prove the existence of real solution(s) to a system of 7 polynomial equalities in 14 variables up to degree 6 for a range of values of 5 real parameters $t_1,t_2,t_3,x_0,x_3$, with $t_1>0,...
0
votes
2answers
35 views

Use Standard Form of line equation to find line using two points?

I can find the equation of a line using the slope-intercept form if I have two points on the line. However I was trying to do the same with the standard form of the equation of a line, $ax+by=c$, and ...
1
vote
3answers
29 views

System of equations has no or infinite solutions?

For which $ \lambda \in \mathbb{R} $ has the following system of equations $ x- \lambda y = 1 $ $ (\lambda - 1)x - 2y = 1 $ a unique solution and no solutions in $ \mathbb{R} $? I solved for $x$ ...
0
votes
1answer
9 views

Forming simultaneous equations tricky wording

Hi this question is in the 'GCSE Mathematics for AQA Higher Student Book' that we use at school I am getting confused with the second simultaneous equation they have derived here as I believe it ...
0
votes
0answers
11 views

Multivariable Equations(s) (both logarithmic and non-logarithmic). Calculating number of roots of equations in each case

Roots and Nature of Roots for Equation(s) in more than 1 Variable For equation in 1 Variable: $x + c = 0$ where $ c \in R $ has exactly one root Similarly, For 2 Variables: $x + y + c = 0$ where $...
1
vote
1answer
33 views

A solution to a transcendental equation

I have a transcendental equation as follows: $$ e^{ax} = b+ax $$ for some constants $a>0$ and $b>0$. Is there a known solution for $x$? I tried on Wolfram and it gave a solution which ...
0
votes
1answer
23 views

System of equations : $1$) $(1+2x)A+(1+2y)B=0 $, $2$) $Ae^{x}+Be^{y}=0$, solving for $A,B$, $x \neq y$ with $x,y$ fixed

As part of some of my DE notes, there is a system of equations where $A,B,x,y \in \Bbb R$, $x \neq y$, $x,y$ fixed. $1$) $(1+2x)A+(1+2y)B=0 $ $2$) $Ae^{x}+Be^{y}=0$ This is all coming from a ...
3
votes
1answer
53 views

For what values of k does this system of equations have a unique / infinite / no solutions?

My system of equations is: \begin{cases} x + 5y- 6z = 2 \\ kx + y - z = 3 \\ 5x - ky + 3z = 7 \end{cases} So the augmented matrix is: $$ \left[ \begin{array}{ccc|c} 1&5&-6&2\\ k&...
0
votes
1answer
15 views

Solving Linear Systems - Criteria for Rank Condition

It is well known that for $A \in \mathbb{K}^{n \times m}$ and $b \in \mathbb{K}^n$ the system $$ Ax = b $$ is solvable if and only if $\operatorname{rank}(A) = \operatorname{rank}(A \mid b)$. I ...
0
votes
2answers
65 views

Solve $(1+c-b)(1+a-c)(1+c-a)=(1+b-c)(1+b-a)(1+a-b)$

I am interested in finding real value solutions to: $(1+c-b)(1+a-c)(1+c-a)=(1+b-c)(1+b-a)(1+a-b)$ Clearly, this is trivially true when $a=b=c$. I'm wondering if there are non-trivial real solutions. ...
1
vote
1answer
36 views

Linear equation with 4 unknowns

I tried to solve these systems of equations in my book: \begin{align} 7&x+4y+3z+2w=46\\ 5&x-y+4w=23\\ &x+z=6\\ 3&x+7w=15 \end{align} I tried to solve it in many different ways, but I ...
0
votes
1answer
25 views

Iinearly independent solutions of homogeneous system of linear equations

Let's take an example of system of equations(actually they are all same) x+2y+4z=0 2x+4y+8z=0 -3x-6y-12z=0 It is homogeneous system of linear equations. Therefore it has zero solution [0,0,0]. ...
47
votes
5answers
3k views

Solution to the equation of a polynomial raised to the power of a polynomial.

The problem at hand is, find the solutions of $x$ in the following equation: $$ (x^2−7x+11)^{x^2−7x+6}=1 $$ My friend who gave me this questions, told me that you can find $6$ solutions without ...
0
votes
0answers
19 views

Steady state solution using time derivatives set to zero

I asked to get the SS solution for non-linear parabolic PDE system: $$\begin{align} \frac{\partial a}{\partial t}&=D \frac{\partial^2 a}{\partial r^2}+f(a,b)\\ \frac{\partial b}{\partial t}&=...
0
votes
1answer
41 views

Search for a constant Metric $\eta$ satisfying the a condition

I am looking for a $2 \times 2$ constant metric $\eta$, satisfying the following condition: $\eta A \eta^{-1}= A^T$. where $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ ...
0
votes
3answers
34 views

Linear system resolution

I've got this linear system \begin{cases} x-3y+6z=0 \\ kx-3y+6z=0\end{cases} I must to find the basis and the size of this linear system. I have a stuck becose here are $kx$. I made it so: $$\begin{...
1
vote
1answer
46 views

How to determine “convexity” in Phase Diagrams for systems of differential equations

Suppose we have a system of differential equations as follows: $$dx/dt = x^2 \\ dy/dt = -y$$ I am certain on how to draw the phase diagram including equilibrium points, isoclines and the general ...