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.

A system of equations is a collection of two or more equations with a same set of unknowns. In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system.

The equations in the system can be linear or non-linear. Linear equations (ones that graph as straight lines) are simpler than non-linear equations, and the simplest linear system is one with two equations and two variables.

Applications:

In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics.

A system of non-linear equations can often be approximated by a linear system, a helpful technique when making a mathematical model or computer simulation of a relatively complex system.

Other tags in conjunction with this tag should specify, whether the equations of the system are linear, polynomial, ordinary or partial differential equations (or something else). This tag has not fully matured yet. See this meta thread for more opinions and discussion.

7557 questions
59
votes
15 answers

Is it possible to have three real numbers that have both their sum and product equal to $1$?

I have to solve $ x+y+z=1$ and $xyz=1$ for a set of $(x, y, z)$. Are there any such real numbers? Edit : What if $x+y+z=xyz=r$, $r$ being an arbitrary real number. Will it still be possible to find real $x$, $y$, $z$?
51
votes
5 answers

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 needing to graph the equation. My approach was this:…
user271938
49
votes
8 answers

Systems of linear equations: Why does no one plug back in?

When someone wants to solve a system of linear equations like $$\begin{cases} 2x+y=0 \\ 3x+y=4 \end{cases}\,,$$ they might use this logic: $$\begin{align} \begin{cases} 2x+y=0 \\ 3x+y=4 \end{cases} \iff &\begin{cases} -2x-y=0 \\ 3x+y=4 \end{cases}…
48
votes
7 answers

How can I prove that 3 planes are arranged in a triangle-like shape without calculating their intersection lines?

The problem So recently in school, we should do a task somewhat like this (roughly translated): Assign a system of linear equations to each drawing Then, there were some systems of three linear equations (SLEs) where each equation was describing a…
Jonas
  • 643
  • 5
  • 18
48
votes
1 answer

Hahn-Banach From Systems of Linear Equations

In this paper1 on the history of functional analysis, the author mentions the following example of an infinite system of linear equations in an infinite number of variables $c_i = A_{ij} x_j$: \begin{align*} \begin{array}{ccccccccc} 1 & = & x_1 & +…
44
votes
6 answers

Find $xy+yz+zx$ given systems of three homogenous quadratic equations for $x, y, z$

This is a question from Math Olympiad. If $\{x,y,z\}\subset\Bbb{R}^+$ and if $$x^2 + xy + y^2 = 3 \\ y^2 + yz + z^2 = 1 \\ x^2 + xz + z^2 = 4$$ find the value of $xy+yz+zx$. I basically do not know how to approach this question. Please let me know…
41
votes
2 answers

On Ramanujan's Question 359

In JIMS 4, p.78, Question 359 was asked by Ramanujan. (See The Problems Submitted by Ramanujan to the Journal of the Indian Mathematical Society, p. 9, by Bruce Berndt, et al.) If, $$\sin(x+y) = 2\sin\big(\tfrac{1}{2}(x-y)\big)\tag1$$ $$\sin(y+z) =…
Tito Piezas III
  • 47,981
  • 5
  • 96
  • 237
37
votes
4 answers

Help with using the Runge-Kutta 4th order method on a system of 2 first order ODE's.

The original ODE I had was $$ \frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0$$ with $y(0)=3$ and $y'(0)=1$. Now I can solve this by hand and obtain that $y(1) = 14.82789927$. However I wish to use the 4th order Runge-Kutta method, so I have the…
37
votes
8 answers

How does Cramer's rule work?

I know Cramer's rule works for 3 linear equations. I know all steps to get solutions. But I don't know why (how) Cramer's rule gives us solutions? Why do we get $x=\frac{\Delta_1}\Delta$ and $y$ and $z$ in the same way? I want to know how these…
Fawad
  • 1,974
  • 3
  • 18
  • 38
32
votes
14 answers

Are there any other methods to apply to solving simultaneous equations?

We are asked to solve for $x$ and $y$ in the following pair of simultaneous equations: $$\begin{align}3x+2y&=36 \tag1\\ 5x+4y&=64\tag2\end{align}$$ I can multiply $(1)$ by $2$, yielding $6x + 4y = 72$, and subtracting $(2)$ from this new equation…
Mr Pie
  • 9,157
  • 3
  • 21
  • 58
32
votes
4 answers

Super hard system of equations

Solve the system of equation for real numbers \begin{split} (a+b) &(c+d) &= 1 & \qquad (1)\\ (a^2+b^2)&(c^2+d^2) &= 9 & \qquad (2)\\ (a^3+b^3)&(c^3+d^3) &= 7 & \qquad (3)\\ (a^4+b^4)&(c^4+d^4) &=25 & \qquad (4)\\ \end{split} First I used…
30
votes
5 answers

System of 4 tedious nonlinear equations: $ (a+k)(b+k)(c+k)(d+k) = $ constant for $1 \le k \le 4$

It is given that $$(a+1)(b+1)(c+1)(d+1)=15$$$$(a+2)(b+2)(c+2)(d+2)=45$$$$(a+3)(b+3)(c+3)(d+3)=133$$$$(a+4)(b+4)(c+4)(d+4)=339$$ How do I find the value of $(a+5)(b+5)(c+5)(d+5)$. I could think only of opening each expression and then manipulating,…
user167045
29
votes
10 answers

What makes a linear system of equations "unsolvable"?

I've been studying simple systems of equations, so I came up with this example off the top of my head: \begin{cases} x + y + z = 1 \\[4px] x + y + 2z = 3 \\[4px] x + y + 3z = -1 \end{cases} Combining the first two equations yields \begin{gather} z…
JSmith
  • 318
  • 1
  • 3
  • 8
27
votes
4 answers

Balance chemical equations without trial and error?

In my AP chemistry class, I often have to balance chemical equations like the following: $$ \mathrm{Al} + \text O_2 \to \mathrm{Al}_2 \mathrm O_3 $$ The goal is to make both side of the arrow have the same amount of atoms by adding compounds in the…
Chao Xu
  • 5,528
  • 3
  • 33
  • 47
27
votes
3 answers

Difference between least squares and minimum norm solution

Consider a linear system of equations $Ax = b$. If the system is overdetermined, the least squares (approximate) solution minimizes $||b - Ax||^2$. Some source sources also mention $||b - Ax||$. If the system is underdetermined one can calculate…
plasmacel
  • 1,142
  • 1
  • 12
  • 26
1
2 3
99 100