# What is System of equations: Definition and 282 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. ### 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'=...
2. ### 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...
3. ### 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...
4. ### 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...
5. ### 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...

7. ### 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 $x_2'+x_2+4x=\sin t$...
8. ### 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..
9. ### 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🕶
10. ### 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.
11. ### 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...
12. ### 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...
13. ### 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 $A^{-1}$
14. ### 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)...
15. ### 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...
16. ### 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
17. ### 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 $x^2+y^2+x+y+xy=4$.
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### 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...
19. ### MHB Integer solutions of system of equations

Find all integer solutions of the system of equations $x+y+z=3$ and $x^3+y^3+z^3=3$.
20. ### 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*}
21. ### 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...
22. ### MHB System of Equations: Find Triples $(x,y,z)$

Find all triples $(x,\,y,\,z)$ of real numbers that satisfy the system of equations $x^3=3x-12y+50\\y^3=12y+3z-2\\z^3=27z+27x$
23. ### 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 \\...
24. ### 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...
25. ### 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##...
26. ### 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\...
27. ### 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...
28. ### 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...
29. ### 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...
30. ### 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...
31. ### 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...
32. ### 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 -...
33. ### 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...
34. ### Shock wave system of equations

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...

43. ### 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...
44. 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...
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### 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...
46. ### 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 =...
47. ### 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...
48. ### A Is this system of equations (numerically) solvable?

Hi, In a project of mine I've encountered the following set of equations: $$\sum_{i=1}^N \left(\frac{1}{M}\sum_{\alpha=1}^Mg_{ij}^\alpha - u_{ij}^* \right) = 0 \qquad \forall: 1\leq j \leq N$$ \sum_{i<j}\left( (u_{ij}^*)^2 - \frac{2}{M^2}\sum_{\alpha < \beta}^Mg_{ij}^\alpha g_{ij}^\beta...
49. ### A How can I find the collision time of 2 ellipsoids that rotate

I have 2 ellipsoids: Ax^2/a^2+Ay^2/b^2+Az^2/c^2=1; (*a,b,c>0 constants*) Bx^2/a^2+By^2/b^2+Bz^2/c^2=1; Ellipsoid A rotates around axis [wax;way;waz] (unit vector) with an infinite speed; Ellipsoid B rotates around axis [wbx;wby;wbz] (unit vector) with an infinite speed; [Dx;Dy;Dz] is the vector...
50. ### Mathematica Solving system of equations Mathematica

I am solving the following system of equations (in matrix form): ##\begin{pmatrix} 1 & -2 & -1 & 1 \\ 2 & -3 & 1 & 6 \\ 3 & -5 & 0 & 7 \\ 1 & 0 & 5 & 9 \end{pmatrix}## I want to solve it using Mathematica, but when I use the command LinearSolve[], I only get back ##\begin{pmatrix} 9 \\ 4 \\ 0...