System of equations Definition and 285 Threads

In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single equations, namely as a:

System of linear equations,
System of nonlinear equations,
System of bilinear equations,
System of polynomial equations,
System of differential equations, or a
System of difference equations

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  1. Lotto

    How to find a system of equations when the solution is given?

    TL;DR Summary: I have to find a system of equations with this solution ## {(1,2,0,3)^T+t(1,1,1,-2)^T+s(1,-1,3,0)^T;s,t \in \mathbb{R}} ## when we know that matrix of this equation has: 1. two non-zero rows 2. 3 non-zero rows. My idea is that I could somehow use the fact that...
  2. Z

    Why can't a critical point of a system of DEs be complex?

    From the first equation we can write $$y=\frac{x}{2}+\frac{x^2}{8}$$ Subbing into the rhs of the second equation and equating to zero we find (after some algebra) that $$x(x-4)(x^2+12x+72)=0$$ This equation has roots ##0##, ##4##, and ##-6\pm 6i##. Then, ##x=0\implies y=0## and...
  3. Z

    How to draw phase portrait for 2x2 nonlinear system of DE?

    The critical points are ##(0,0)## and ##(2,1)##. The linearization of these equations is $$\begin{bmatrix}x'\\y'\end{bmatrix}=\begin{bmatrix}-1+y_0&x_0\\y_0&x_0-2\end{bmatrix}\begin{bmatrix}x-x_0\\y-y_0\end{bmatrix}$$ At ##(0,0)## we have $$\begin{bmatrix}x'\\...
  4. berlinvic

    Solving system of differential equations using elimination method

    I am trying to solve this system of differential equations using elimination method, but I am stuck. $$\begin{cases} y'_1 = y_2, \\ y'_2 = -y_1 + \frac{1}{\cos x} \end{cases}$$ Here's what I tried: I've been suggested to differentiate the ##y_1'= y_2## again to get ##y_1''= y_2'=...
  5. M

    Finding the solution to system of 3 equations with 3 unknowns

    For this, I am trying to find solutions, however, I think I am getting a strange result that I am not too sure how to intercept. I first multiply the first equation by 2 to get ##2x_1 - 8x_3 = 4## and then I add it to the second equation below to get ##0 = 1##. I think this means that there...
  6. C

    How Can I Solve a System of Equations With Complex Numbers?

    How can I solve a system of equations with complex numbers 2z+w=7i zi+w=-1 I have tried substituting z with a+bi and I have tried substituting w=7i-2z but didn't get anything useful. Edit: also, I've tried, multiplying lower eq. with -1 so that I can cancel w but I get stuck with 2z and zi and...
  7. A

    Can this system of inequalities be solved for x?

    Summary: Can these two equations be solved for x like a system of linear inequalities, and how? ##x- 2y \le 54## ##x + y \ge 93## We start with ##x- 2y \le 54## ##x + y \ge 93## Multiplying the second equation by 2, we have ##2x + 2y \ge 184##. We cannot seem to cancel the y out with the...
  8. B

    MHB Number of solutions for system of equations

    Hello! I have a simple question about solutions, better said number of solutions for this system of equations. \[ \begin{cases} x_{1 } − x_{2 } + 3x_{3 } − 2x_{4 } = 1\\ −2x_{1 } + 2cx_{2 } − 4x_{3 } + 2x_{4 } = −7\\ − 2x_{3 } + (−c + 6)x_{4 } = 2c + 15\\ − 2x_{3 } + c^{2 }x_{4 } = c^{2...
  9. B

    MHB Solve System of Equations Related to Race Speeds

  10. karush

    MHB Solving 2nd-Order IVP as System of Equations

    $\tiny{2.1.5.1.c}$ source Change the second-order IVP into a system of equations $\dfrac{d^2x}{dt^2}+\dfrac{dx}{dt}'+4x=\sin t \quad x(0)=4\quad x'(0)= -3$ ok I presume we can rewrite this as $u''+u'+4u=\sin t$ Let $x_1=u$ and $x_2=u'$ then $x_1'=x_2$ substituting $x_2'+x_2+4x=\sin t$...
  11. karush

    MHB System of Equations for Second-Order IVP

    Change the second-order IVP into a system of equations $y''+y'-2y=0\quad y(0)= 2\quad y'(0)=0$ let $x_1=y$ and $x_2=y'$ then $x_1'= x_2$ and $y''=x_2'$ then by substitution $x_2'+x_2-2x_1=0$ then the system of first order of equations $x_1'=x_2$ $x_2'=-x_2+2x_1$ hopefully so far..
  12. karush

    MHB 097 Change the second-order IVP into a system of equations

    $\tiny{2.1.5.1}$ Change the second-order initial-value problem into a system of equations $x''+6x'-2x= 0\quad x(0)=1\quad x'(0)=1$ ok my first step was to do this $e^{rt}(r^2+6r-2)=0$ using quadratic formula we get $r=-3+\sqrt{11},\quad r=-3-\sqrt{11}$ just seeing if I going down the right road🕶
  13. anemone

    MHB Solutions of System of Equations

    Find all solutions of the system of equations $s(x)+s(y)=x\\ x+y+s(z)=z\\ s(x)+s(y)+s(z)=y-4$ where $x,\,y$ and $z$ are positive integers, and $s(x),\,s(y)$ and $s(z)$ are the numbers of digits in the decimal representations of $x,\,y$ and $z$ respectively.
  14. greg_rack

    Solving System of Equations: Understanding the Analytical Reasons

    Hi guys, I managed to solve this problem just by "rewriting" the first equation of the system as ##t=f(x)## and then substituting that in the second ##y=f(t)## equation, ending(of course) up with the sought ##f(x,y)## function. The problem here is I didn't really understand what I have done and...
  15. K

    System of equations and solving for an unknown

    The first thing I do is making the argumented matrix: Then I try to rearrange to make the row echelon form. But maybe that's what confusses me the most. I have tried different ways of doing it, for example changing the order of the equations. I always end up with ##k+number## expression in...
  16. karush

    MHB 311.2.2.6 use inverse matrix to solve system of equations

    $\tiny{311.2.2.6}$ Use the inverse to solve the system $\begin{array}{rrrrr} 7x_1&+3x_2&=-9\\ -2x_1&+x_2&=10 \end{array}$ the thing I could not get here without a calculator is $A^{-1}$
  17. M

    MHB Linear system of equations: Echelon form/Solutions

    Hey! 😊 I am looking at the following exercise but I think that I miss something. The statement is the following: We are given the following system of equations: \begin{align*}2a-2c+d-2e=&-2 \\ -2c-2d+2e=&\ \ \ \ \ 3 \\ d+2e=&-2\end{align*} 1) Is the system in echelon form? Justify. 2)...
  18. T

    Finding a Unique Solution to a System of Equations

    It makes sense that a=2 would cause problems because then we wouldn't have a matrix of full rank and we'd be unable to determine a value for w. But the key also says that when b+4a^2-4a-7≠0. Why is that an issue? For example, if a=1, that just says implies that w=0. Through back-subsitution...
  19. S

    MHB Solving System of Equations: xy, yz, zx

    Solve the following systemof equations: $\dfrac{xy}{x+y}=a$ $\dfrac{yz}{y+z}=b$ $\dfrac{zx}{z+x}=c$ where a,b,c are not zero
  20. anemone

    MHB Solving $x^3+y^3=7$ and $x^2+y^2+x+y+xy=4$ System of Equations

    Find all real $x$ and $y$ that satisfy the system $x^3+y^3=7$ and $x^2+y^2+x+y+xy=4$.
  21. M

    Mathematica Solve a system of equations numerically

    Hi PF! I'm trying to solve three equations in Mathematica, but NSolve is taking FOREVER. Am I missing perhaps an easier way? The equations are below: NSolve[{1/2 r (r \[Theta] + (2 h + r Cos[\[Theta]]) Sin[\[Theta]]) == v, Cos[\[Alpha]] == -Sin[\[Theta] - \[Beta]], Tan[\[Theta]] == (-h...
  22. anemone

    MHB Integer solutions of system of equations

    Find all integer solutions of the system of equations $x+y+z=3$ and $x^3+y^3+z^3=3$.
  23. anemone

    MHB Real Numbers $a,\,b,\,c$ Solving System of Equations

    Find all the real numbers $a,\,b$ and $c$ that satisfy the following system of equations: $\begin{align*}a + b + c &= 1\\ \dfrac{a}{ 1 - a}+\dfrac{b}{1 - b} + \dfrac{c}{1 - c} &= 6ac + 6bc = (a + 1)(b + 1)\end{align*}$
  24. LCSphysicist

    Analysing System of Equations: 2kx2 + kx1 = mx2 & 2kx1 + kx2 + kXocos(wt) = mx1

    Well, i think the important here is the system, what you think about?: -2kx2 + kx1 = mx2'' -2kx1 + kx2 + kXocos(wt) = mx1'' After this, is just solve, i found: x2 = (k*xo*cos(wt)*(4k/m - 2w²))/(2m*(k/m - w²)*(3k/m - w²)) The cool is that if we put w equal the two normal frequency x2 tends to...
  25. anemone

    MHB System of Equations: Find Triples $(x,y,z)$

    Find all triples $(x,\,y,\,z)$ of real numbers that satisfy the system of equations $x^3=3x-12y+50\\y^3=12y+3z-2\\z^3=27z+27x$
  26. D

    Question about the solution of this system of equations

    hi given such system of equations ## \begin{cases} \rho^2 = 2 \rho \\ 2\theta= -\theta+2k\pi , k\in \mathbb Z \\ \end{cases} ## in the solution of the professor the system is solved is solved as follows. ## \begin{cases} \rho=0 , \rho=2 \\ \theta= -\frac 2 3 k\pi , k = 0,1,2 \\...
  27. H

    I Solving this system of equations in different ways

    Good night! How do I find the values of a (real) so that the solution of this system is? (i) just an ordered pair? (ii) exactly two pairs. (iii) exactly 3? (iv) is there a place where you have more than 3 pairs as an answer?So... I thought like this: I developed the first part. I solved the...
  28. D

    I Logarithmic terms in a system of equations

    (I hope this is not a double posting) I want to solve this system of equations, containing logarithmic terms: ##7\ln(a/b)+A = 7\ln(d/e)+D = 7\ln(g/h)+G## ##7\ln(a/c)+B = 7\ln(d/f)+E = 7\ln(g/i)+H## ##7\ln(b/c)+C = 7\ln(e/f)+F = 7\ln(h/i)+I## ##a\phi_1+d\phi_2+g\phi_3=X##...
  29. H

    I Building a coefficient matrix for a system of equations

    I want to solve the following system of equations ##M_{1} = f_1+f_2+m_1+m_2\ \ ;\ \ M_{7} = f_1+f_2+s_1+s_2\ \ ;\ \ M_{13} = m_1+m_2+s_1+s_2## ##M_{2} = f_1+f_3+m_1+m_3\ \ ;\ \ M_{8} = f_1+f_3+s_1+s_3\ \ ;\ \ M_{14} = m_1+m_3+s_1+s_3## ##M_{3} = f_1+f_4+m_1+m_4\ \ ;\ \ M_{9} = f_1+f_4+s_1+s_4\...
  30. Mina Farag

    Applying the implicit function theorem to a system of equations

    My attempt: According to the implicit function theorem as long as the determinant of the jacobian given by ∂(F,G)/∂(y,z) is not equal to 0, the parametrization is possible. ∂(F,G)/∂(y,z)=4yzMeaning that all points where z and y are not equal to 0 are possible parametrizations. My friend's...
  31. currently

    Deriving the first-order system for this governing equation

    I tried finding the solution of the equation itself but it hasn't helped! Links to concepts would be greatly appreciated...thank you...
  32. H

    A Diffusion equation and a system of equations with reciprocal unknowns?

    So the normal diffusion equation looks like \frac{\partial c}{\partial t} = k\frac{\partial}{\partial x}\left(\frac{\partial c}{\partial x}\right) I know how to get explicit and implicit solutions to this equation using finite differences. However, I am trying to do the same for an equation of...
  33. S

    How Do You Solve Systems of Linear Equations with Parameters?

    1) x = 3 - 4p + q x = 3 - 4y + z x + 4y - z = 3 2) x + 4y - z = 3 (i) let x = a and y = b, so z = a + 4b - 3 General solution: x = a y = b z = a+ 4b - 3 (ii) let x = r and z = t, so y = (3 - r + t) / 4 General solution: x = r y = (3 - r + t) / 4 z = t3) I don't understand this part. Is the...
  34. K

    I Find Practical Resonance Frequencies in Linear Differential Equations

    Hi all, I would like to know what is the equation upon which I can use to determine the practical resonance frequencies in a system of second order, linear differential equations. First some definitions: What I mean by practical resonance frequencies, is the frequencies that a second order...
  35. EEristavi

    Solving a System of Equations via the Matrix Method

    I have equation system: x + y + z - a*k = 0 -b*x + y + z = 0 -c*y + z = 0 -d*x + y = 0 where: a, b, c, d = const. Have to find: x, y, z, k Attempt of solution: I create Matrix A with coefficients; Matrix B - Solutions (Zeros) and Matrix X - variables. When I try to use Cramer's rule -...
  36. X

    A Need help solving Darboux equation

    I'm working on a personal math project and I'm running into this system of differential equations. I have seen references which state the solutions are in terms of Hermite modular elliptic functions, but I do not know what those functions are. All of the references I can find on this equation...
  37. F

    Struggling with Rankine-Hugoniot Conditions?

    Homework Statement This is Rankine-Hugoniot conditions at a hydrodynamic shock front. Where P2=0 v2=0. The problem is attached. I need to solve a system of equations. I thought it would be relatively straight forward solving for the three unknowns but I'm struggling. I know it's possible to...
  38. M

    Problem involving a system of equations

    Homework Statement If an amount of $1000 is deposited in a savings account that pays 3.2% interest per year compounded monthly, the amount in the account after nmonths is given by: The amount in the account after 2 years (rounded to one decimal point) will be??
  39. M

    Why do extra moment equations not count toward a system of equations?

    I was just looking at indeterminate statics problems where you have a beam, three elastic wires (left, centre, and right that are holding it up), and some extra mass. (Just like example 5 in the notes below: http://fast10.vsb.cz/lausova/indeterm_all.pdf). I understand the method that was used to...
  40. C

    Solving System of Equations (including Trig)

    Homework Statement This is an example worked out in the textbook Matter and Interactions, 4th Edition (pg. 181). The authors assume that solving for two unknowns is no problem, so they don’t show the steps. I’m trying to work it out and am stuck. I’ve used Alpha to get a step by step solution...
  41. opus

    B Can Linear Systems of Equations Have More Than Three Solutions?

    According to my text, a linear system of equations is a problem described by two or more equations in two or more variables. Now the individual equations have infinitely many solutions, however, the system of equations is said to have either exactly one solution (one point of intersection...
  42. Jayalk97

    Linearizing a system of equations?

    Homework Statement So these are the equations of motion for a quad-copter. I am supposed to create a MATLAB model for the z-axis. In order to do this I have to linearize the equations around these points, and arrange them in state space representation. Homework Equations As above The...
  43. R

    Creating system of equations from word problem optimization

    I have this word problem, and was wondering how I would go about creating a system of equations. Here is the question: Problem: You are a small forest landowner, and decide you want to sustainably harvest some of timber on your property. There are costs related to the infrastructure needed to...
  44. P

    MHB Mason's question via Facebook about solving a system of equations (2)

    As all the z coefficients are the same, it's a good idea to eliminate the z values in the second and third equations, so apply R2 - R1 to R2 and R3 - R1 to R3... $\displaystyle \begin{align*} z &= 12 - x + 4\,y \\ 0 &= -8 + 6\,x - y \\ 0 &= -7 - 11\,x - 9\,y \end{align*}$ Now we can multiply...
  45. P

    MHB Mason's question via Facebook about solving a system of equations

    The LCM of the $\displaystyle \begin{align*} x \end{align*}$ coefficients is 30, so multiplying the first equation by 6, the second by 10 and the third by 5 gives $\displaystyle \begin{align*} 30\,x - 12\,y + 6\,z &= 18 \\ 30\,x + 10\,y + 30\,z &= 50 \\ 30\,x + 5\,y - 20\,z &= 310...
  46. M

    MHB System of equations - Relative error

    We have the linear system of equations $Ax=b$ with \begin{equation*}A=\begin{pmatrix}0 & 1 & 1 \\ 0.5 & 1.0001 & 3 \\ 1 & 2 & 4\end{pmatrix} \ \ \ \text{ und } \ \ \ b=\begin{pmatrix}2 \\ 3 \\ 4\end{pmatrix}\end{equation*} First, I want to calculate the solution using the Gauss algorithm with...
  47. M

    Solve the system of equations?

    Homework Statement Solve the system of equations x1-3x2-2x3=0 -x1+2x2+x3=0 2x1+4x2+6x3=0 using either Gaussian or Gauss-Jordan elimination. Homework Equations None. The Attempt at a Solution R1+R2, I got x1-3x2-2x3=0 -x2-x3=0 2x1+4x2+6x3=0...
  48. M

    Finding values to make a linear system consistent

    Homework Statement Given the following matrix: I need to determine the conditions for b1, b2, and b3 to make the system consistent. In addition, I need to check if the system is consistent when: a) b1 = 1, b2 = 1, b3 = 3 b) b1 = 1, b2 = 0., b3 = -1 c) b1 = 1, b2 = 2, b3 = 3 Homework...
  49. Oannes

    I Solving System of Equations w/ Gauss-Jordan Elimination

    I am fairly new here so I apologize for any mistakes in my post. My question concerning solving a system of equations using Gauss-Jordan Elimination is specifically about different ways to handle a possible constant. Say for instance you have three equations: X1+X2+X3 + 3 = 9 2X1+4X2+X3 =...
  50. T

    A Getting as close as possible to a solution (system of equations)?

    Hi, I have a set of equations that look like this: y1 = k1*x1 + k2*x2 - A1 = 0 y2 = k3*x1 + k4*x3 - A2 = 0 y3 = k5*x2 + k6*x4 - A3 = 0 y4 = k7*x3 - A4 = 0 y5 = k8*x4 - A5 = 0 k1 to k8 are known positive constants. A1 to A5 are known positive constants (I will use...
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