What is Systems of equations: Definition and 58 Discussions

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. M

    Find the solutions of the system, for all λ

    Thread moved from technical math section, so there is no homework template. (∀λ∃ℝ) -x + y - z = 1 -2x + 10y + (2λ + 6) = 6 3x + 11y + (λ2+6)z = 5λ - 1 after gaussian elimination I have this: -1 4 -2 | 1 0 1 λ | 2 0 0 λ(λ-1) | 5λ So, for λ=0 ⇒ ∞ solutions, for λ=1...
  2. C

    Using inverses to solve systems of equations

    For this, Can someone please tell me where they got ##X = IX## from? Many thanks!
  3. E

    I Fundamental matrix of a second order 2x2 system of ODEs

    Let ## \mathbf{x''} = A\mathbf{x} ## be a homogenous second order system of linear differential equations where ## A = \begin{bmatrix} a & b\\ c & d \end{bmatrix} ## and ## \mathbf{x} = \begin{bmatrix} x(t)\\ y(t)) \end{bmatrix} ## Now to solve this equation we transform it into a 4x4...
  4. K

    B Solutions to systems of equations

    The set of equations $$A = a_1 x + a_2y$$ ¨$$B = b_1 x + b_2 y$$ can be solved for the unkowns ##x## and ##y##. Does it make sense to have ##x## and ##y## be known and instead ##A## and ##B## unkown?
  5. R

    Solving systems of equations using Jordan exchanges

    2u + 3v + 3w = 2 + 5v + 7w = 2 6u + 9v + 8w = 5 ##\begin{bmatrix} 2 & 3 & 3 & 2 \\ 0 & 5 & 7 & 2 \\ 6 & 9 & 8 & 5 \end{bmatrix}## We have been asked to use Jordan exchange to solve the above equations. Can someone please explain how to determine the values for r, s for the equations...
  6. Telemachus

    I Vector that satisfies systems of equations simultaneously

    Hi. I want to normalize a discretized function ##p_{k,k'}##, to satisfy simultaneously two conditions. The normalized function ##p^*_{k,k'}## has to satisfy simultaneously: 1) ##\sum_{k=1}^{M} p^*_{k,k'} w_k=1##, for all ##k'=1,2,...,M##; 2) ##\sum_{k=1}^{M} p^*_{k,k'} w_k \hat \Omega_k \cdot...
  7. M

    Constructing a 3x3 Linear system question

    Homework Statement Construct a 3 × 3 example of a linear system that has 9 different coefficients on the left hand side but rows 2 and 3 become zero in elimination. If the right hand sude of your system is <b1,b2,b3> (Imagine this is a column vector) then how many solutions does your system...
  8. F

    I How do you solve systems of equations in complex physics problems?

    It seems like systems of equation are part of almost every complex physics problem. What exactly are the rules or steps for solving such equations? I know how to solve a basic linear system of equations with two variables, but how do solve more complicated systems like non-linear systems of...
  9. nightingale123

    What Values of ##\lambda## Allow Non-Trivial Solutions in This Linear System?

    Homework Statement 3.For which values of ##\lambda## does the following system of equations also have non trivial solutions Homework EquationsThe Attempt at a Solution What I tried doing first is to put all variables on the same side and got ## v+y-\lambda*x=0\\ x+z-\lambda*y=0\\...
  10. Cocoleia

    Solving systems of equations that contain complex numbers

    Homework Statement I am having trouble solving systems of equations when they contain complex numbers. The context is circuit theory and phasors. For example, I am given this And the goal is to find I2 and Voc, which you can see the answers for. I just don't know how to manipulate the numbers...
  11. P

    B Why is there a solution to this system?

    Even though school's been over since last week, I've set aside this problem because I cannot figure out how it works and the teacher only posted the solution (without the steps) on my blackboard. So, here's a system of equations that I had to solve: ##10x+24y+2z=-18## ##-2x-7y+4z=6##...
  12. faradayscat

    I Non-homogeneous systems with repeated eigenvalues

    Quick question, can you solve non-homogeneous systems with repeated eigenvalues the same ways? i.e. variation of parameters, undetermined coefficients, etc... would the fundamental matrix contain the solution with the generalized eigenvalue? Thanks!
  13. Y

    MHB Solving Systems of Equations with Whole Numbers

    Hello all I have a couple of short questions, both similar, which I do not know how to even start, and I could use some help. 1) Show that if in the system Ax=b, det(A)=-1, and all the members of A are whole numbers (belong to Z), and all the members of b are whole numbers (belong to Z), then...
  14. Estanho

    Transform that maps points from any quad to an reactangle

    Hello, This question might seem silly, but I've tried some approaches and none of them seemed to work. Here's my problem: I need some sort of transform that maps points from any quad to an rectangle. I will be using this on a computer graphics software, so you can think of this rectangle as my...
  15. JR Sauerland

    Systems of Equations: What's your favorite way to solve them?

    As the title suggests, I want the tips from the pros. I've learned all of the methods, but going into Calculus/Physics, which methods should I take preference to? Furthermore, will there be significant use of systems of equations in Physics or Calculus in college? P.S.: I know how to graph the...
  16. D

    How to solve a very large overdetermined system numerically?

    I am doing a project on image processing and I need to solve the following set of equations: nx+nz*( z(x+1,y)-z(x,y) )=0 ny+nz*( z(x+1,y)-z(x,y) )=0 and equations of the boundary (bottom and right side of the image): nx+nz*( z(x,y)-z(x-1,y) )=0 ny+nz*( z(x,y)-z(x,y-1) )=0 nx,ny,nz is the...
  17. C

    Compound Interest Systems of Equations Problem

    Homework Statement Romeo was given a gift of $10,000 when he turned 16. He invested it at 3% per annum. Three years later, Juliet was given $10,000, which she invested at 5% per annum. When will the two amounts be equal in value? Homework Equations Compound Interest Formula Total =...
  18. Avatrin

    Matrices and systems of equations

    Hi I am now filling in what I perceive to be gaps in my knowledge. One of these problems is understanding why matrices can solve systems of equations. I do completely get Gaussian elimination to solve systems of linear equations. However, when using determinants and the like to solve, for...
  19. I

    Systems of Equations word problems

    Hi, I have three systems of equations word problems. Can you help me? #1: The sum of three numbers is 32. The largest if 4 times the smallest. The sum of the two smaller numbers is 8 less than the largest. What are the numbers? I have tried to model this one. I got: _ + _ +_ = 32 Largest = 4S...
  20. I

    Systems of Equations with Complex Numbers

    Mod note: This thread was moved from a technical math section, so doesn't include the homework template. I know this has been asked before, but none of the other posts have helped me. I cannot for the life of me figure out how to solve a system of equations with complex numbers. Here is a very...
  21. Aristotle

    How Do You Calculate the Preimage of a Vector Under a Linear Transformation?

    Consider the linear transformation T: R3 --> R3 /w T(v1,v2,v3)=(0, v1+v2, v2+v3) What is the preimage of w=(0,2,5) ?I tried setting up the system of equations and got v1+v2= 2 and v2+v3=5 but after that I got kinda lost in how to find the individual solutions?
  22. T

    Lagrange multiplier systems of equations -- Help please

    Homework Statement Hi guys I am new here and i really need help with this question. I've tried it multiple times but can't find all the critical points, help would be greatly appreciated. the question is as follows: Find the maximum and minimum values of w=4x-(1/2)y+(27/2)z on the surface...
  23. I

    Finding the value of a variable in a matrix

    Homework Statement Given :  x+y+5z = 2   x+2y+7z = 1   2x−y+4z = a . a) Determine the value of a which will make the given system have many solutions. Explain your answer. b) Choose a value of a which will make the given system have NO solutions. Explain your answer. c) Is it possible...
  24. R

    Linear Algebra: Solving a system with free variables

    Here is a picture of the problem. Can anyone give me some hints on the problem? I've looked in my textbook, but I don't know what "s" means. I found stuff on the parametric vector form, and it gives me the equation x = su + tv, but I don't see any "t"'s in this problem. I first tried...
  25. L

    Solve Systems of Equations 17.3-(17.3)e^(-92.34940680845194549420x)^y

    17.3-(17.3)e^(-92.34940680845194549420x)^y)=17.30181504460159157646-((17.3-(17.3)e^((-0.00118329948908244714x)^y)) anyone know how to solve this or a calculator that can?
  26. C

    Systems of equations with many equations, more than 3

    I'm doing quadratics and I broke down some data into 8 systems of equations and I was wondering what the best way to solve this was. Someone online mentioned doing Gaussian Matrix but I don't know what that is. Any suggestions?
  27. Y

    MHB Systems of equations - further understanding

    Hello again, I have a few more questions regarding systems of equations, I will collect them all here in one post since they are small. 1. The first is the following system: x+2y-3z=a 3x-y+2z=b x-5y+8z=c I need to determine the relation between a,b and c for which the system has infinite...
  28. C

    Solving systems of equations with trig functions

    I've stumbled upon a system of equations that involves trig functions... 100cos(θ) + 200cos(ω) = 250 100sin(θ) + 200sin(ω) = 0 How do you go about solving a system like this? It's nonlinear, so linear algebra won't work...
  29. J

    For Statics:Simple Systems of Equations to Solve Equilibrium Problems

    Homework Statement I have a picture attached with this static equilibrium problem. (Solve system for Fac and Fab) Homework Equations Sum of x and y both=0The Attempt at a Solution I just don't have a systematic easy approach to solving these systems. Does anyone have one? From the picture...
  30. Jameson

    MHB Cammie's question from Facebook (finding a mixture through systems of equations)

    A girl scout troop has 20 pounds of candy worth 80 cents per pound. the troop wishes to mix it with candy worth 50 cents per pound so that the total mixture can be sold at 60 cents per pound without any gain or loss. how much of the 50 cent candy must be used? Solve by elimination.
  31. Mandelbroth

    Logarithms in Systems of Equations

    Homework Statement Consider the system... y=e^{-x}+1 \\ y=2+lnx The Attempt at a Solution I don't think there is a "pretty way" (algebraic manipulation) to do this. I simply found the point where the plots intercepted, getting the solution set (0.62745018..., 1.5339409...). Is there a...
  32. K

    Calculators TI-36X Pro solving systems of equations with complex numbers

    Hello all, I was wondering if there was a way of solving a matrix on a TI-36X Pro that has complex numbers in it. Every time I try, it just says "invalid data type". Is there any way of getting around this? Thanks! Robert
  33. T

    MHB Setting up systems of equations

    I have this problem in my book: Ann had 20(Cash) less than Betty. Ann spent 3/4 of her money, while Betty spent 4/5 of hers. Then Ann's remainder was 5/6 of Betty's remainder. If Ann had x(Cash) originally, form an equation in x and solve it. And this solution in the answer key: x=Ann's...
  34. G

    Roots of polynomials as nonlinear systems of equations

    Ok, to start off I have been examining the structure of polynomials. For instance, consider the general polynomial P(x)=\sum^{n}_{k=0}a_{k}x^{k} (1) Given some polynomial, the coefficients are known. Without the loss of generality...
  35. D

    Incredibly simple question about the logic behind systems of equations

    I know this is a simple question but I can't exactly figure out the logic governing this problem, I just know it had to be this way. Let's say I have two equations: x - 4y + 9 = 0 y - 3x + 5 = 0 If I set them equal to each other then I get no where because I'll have both x and y in the...
  36. I

    Good method for checking solutions to linear systems of equations by hand

    Hello all, I was wondering if any of you have a good method for checking your answers to linear systems of equations(when working by hand). I mean the sorts of equations you would encounter on a first university linear algebra course. I solve the system with Gauss/Gauss-Jordan elimination...
  37. R

    A quick and clever method for solving nonlinear systems of equations

    if we are given a polynomial s^6+as^5+bs^4+cs^3+ds^2+es+k (if a,b,c,d are known) what is a clever method to solving for the value k if we are given the following: the above polynomial is equal to the following(zeta is given as some constant, say 1 for simplicity)...
  38. S

    Systems of Equations: Word Problem

    Homework Statement A small plane flying into the wind takes 3 h 20 min to complete a flight of 960km. Flying with the wind, the same plane takes 2 h 30 mins to make the trip. What is the speed of the plane? What is the speed of the wind? Homework Equations N/A The Attempt at a...
  39. N

    Do these 3 systems of equations all all define the same curve?

    Homework Statement Consider three systems of equations: x^2 + y^2 + z^2 = 1 y^2 + z^2 = 1 x^2 + y^2 + z^2 = 1 x = 0 y^2 + z^2 = 1 x = 0 Which of these define the same curve and which define different ones? Homework Equations x^2 + y^2 + z^2 = R^2 is a sphere x,y, or z =...
  40. S

    Systems of equations using partial derivatives

    Homework Statement Consider the system of equations x^2y+za+b^2=1 y^3z+x-ab=0 xb+ya+xyz=-1 1. Can the system be solved for x, y, z as functions of a and b near the point (x, y, z, a, b)=(-1, 1, 1, 0, 0)? 2. Find \frac{\partial x}{\partial a} where x=x(a, b) The Attempt at a...
  41. T

    Linear equation, span, vectors, linear systems of equations

    show that S and T have the same span in R^3 by showing that the vectors in S are in the span of T and vise versa. S= {(1,0,0), (0,1,0)} T= {(1,2,0), (2,1,0)}im a little confused on how to start off on this problem.. help?!
  42. L

    Linear Algebra Systems of Equations

    Homework Statement Consider the following two system of equations: y_{1}=-2x_{1}-x_{2}+2x_{3} y_{2}=2x_{1}+2x_{2}-3x_{3} y_{3}=-2x_{1}-2x_{2}+2x_{3} and z_{1}=3y_{1}-4y_{2}-3y_{3} z_{1}=3y_{1}-y_{2}-4y_{3} Rewrite these 2 systems as Ax=y and By=z. Use this to get C so that Cx=z...
  43. L

    MATLAB Systems of equations in matlab

    hello everybody! T1=A T1'+B(T0'+T2'+T0+T2) T2=A T2'+B(T1'+T3'+T1+T3) T3=A T3'+B(T2'+T4'+T2+T4) if A,B,T0,T4 and everything in (') character is known, i have T1,T2,T3 to find and 3 equations. how do i use MATLAB to solve this?(i have a...
  44. C

    Mathematica Mathematica and systems of equations

    I'd like to write (so not calculate, just "publish") some notes using mathematica. My very big problem in fact is the text-mode... How could I write a system of equations in mathematica, so that it appears "beauty", divided into multiple lines? I don't want something like this: f =...
  45. P

    Solving systems of equations >2 simultaneous

    Homework Statement Hi, I'm doing nodal analysis and I end up with 3-5 equations that need solving simultaneously. I have been sticking them in mathematica and letting it do its thing but I need to be able to do it on my 89 too. The trouble is that there not in a matrix friendly form - - -...
  46. S

    Solving Systems of Equations over F2

    Homework Statement Find all solutions to the system of equations with over the binary field F2 w+x +z=0 x+y =1 enter you answer as a list of point (w,x,y,z) Homework Equations The Attempt at a Solution This question should be easy, but I just don't understand binary field...
  47. T

    Systems Of Equations Word Problems

    Homework Statement 1. A small corporation borrowed $1,500,000 to expand it's product line. Some of the money was borrowed at 8%, some at 9%, and some at 12%. How much was borrowed at each rate if the annual interest was $133,000 and the amount borrowed at 8% was 4 times the amount borrowed at...
  48. R

    Unsolvable Force Components Problem

    Homework Statement I've got this physics problem where I'm working to find the magnitudes of two force components. Homework Equations F1sin45 + F2cos30 = 500 lbf F1sin45 - F2sin30 = 0 The Attempt at a Solution I've tried so many times by substituting F1 and F2 into both equations but I...
  49. J

    Solving systems of equations with complex numbers

    I can easily solve systems of equations with no complex numbers, could somebody give a short overview of the easiest way to do this by hand. thanks
  50. Saladsamurai

    Silly Question about systems of equations

    I had a thought while doing some calculus last night and it now concerns me that for some reason I cannot answer it. If given the equations y=x and y=x^2+1 it would appear that since the range of the latter "reaches" infinity "first" that these two curves never meet. Now that thought did not...