Equation Definition and 140 Discussions

In mathematics, an equation is a statement that asserts the equality of two expressions, which are connected by the equals sign "=". The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any equality is an equation.Solving an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables.An equation is written as two expressions, connected by an equals sign ("="). The expressions on the two sides of the equals sign are called the "left-hand side" and "right-hand side" of the equation. Very often the right-hand side of an equation is assumed to be zero. Assuming this does not reduce the generality, as this can be realized by subtracting the right-hand side from both sides.
The most common type of equation is a polynomial equation (commonly called also an algebraic equation) in which the two sides are polynomials.
The sides of a polynomial equation contain one or more terms. For example, the equation






{\displaystyle Ax^{2}+Bx+C-y=0}
has left-hand side






{\displaystyle Ax^{2}+Bx+C-y}
, which has four terms, and right-hand side


{\displaystyle 0}
, consisting of just one term. The names of the variables suggest that x and y are unknowns, and that A, B, and C are parameters, but this is normally fixed by the context (in some contexts, y may be a parameter, or A, B, and C may be ordinary variables).
An equation is analogous to a scale into which weights are placed. When equal weights of something (e.g., grain) are placed into the two pans, the two weights cause the scale to be in balance and are said to be equal. If a quantity of grain is removed from one pan of the balance, an equal amount of grain must be removed from the other pan to keep the scale in balance. More generally, an equation remains in balance if the same operation is performed on its both sides.
In Cartesian geometry, equations are used to describe geometric figures. As the equations that are considered, such as implicit equations or parametric equations, have infinitely many solutions, the objective is now different: instead of giving the solutions explicitly or counting them, which is impossible, one uses equations for studying properties of figures. This is the starting idea of algebraic geometry, an important area of mathematics.
Algebra studies two main families of equations: polynomial equations and, among them, the special case of linear equations. When there is only one variable, polynomial equations have the form P(x) = 0, where P is a polynomial, and linear equations have the form ax + b = 0, where a and b are parameters. To solve equations from either family, one uses algorithmic or geometric techniques that originate from linear algebra or mathematical analysis. Algebra also studies Diophantine equations where the coefficients and solutions are integers. The techniques used are different and come from number theory. These equations are difficult in general; one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions.
Differential equations are equations that involve one or more functions and their derivatives. They are solved by finding an expression for the function that does not involve derivatives. Differential equations are used to model processes that involve the rates of change of the variable, and are used in areas such as physics, chemistry, biology, and economics.
The "=" symbol, which appears in every equation, was invented in 1557 by Robert Recorde, who considered that nothing could be more equal than parallel straight lines with the same length.

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  1. brotherbobby

    Solving a nested logarithmic equation

    Problem statement : Let me copy and paste the problem on the right as it appears in the text. Solution : Using the Relevant Equations (2) and (3) above, we can claim that \begin{align*} &\log_{2x^2+3x+5}(x^2+3)=1\\ &\Rightarrow x^2+3 = 2x^2+3x+5\\ &\Rightarrow x^2+3x+2=0\\ &\Rightarrow...
  2. N

    Find an equivalent equation involving trig functions

    Rewrite the given equation, attempt 1: ##2\sin(x)\cos(x) + 2\sin(x) + 2\cos(x) = 0## ##\sin(x)\cos(x) + \sin(x) + \cos(x) = 0## ##\sin(x)(\cos(x) + 1) + \cos(x) = 0##, naaah, can't get any relevant out from here. Attempt 2: ##2\sin(x)\cos(x) + 2\sqrt{2}*\sin(x + \pi/4) = 0## ##\sin(x)\cos(x) +...
  3. F

    I Having problems finding an equation for radioisotope production yield

    I am trying to get a plot, or a result for target like this: I've found some equations for my requested results, but they are from the same person's articles and I couldn't find another source. I've searched for EOB and TTY calculations but couldn't really do something. Could you give me an...
  4. NODARman

    I Triple equation for integral on a graph

    Hi, so I'm trying to find the volume of a shape using integral, I found the equation of one plane in 3D space but the second one is something like that, which I cannot write in integral as a function: ##\frac{2(2x-a)}{a}=-\frac{2(6y-a\sqrt3)}{a\sqrt3}=\frac{2z-a\sqrt3}{a\sqrt3}## In the 3D...
  5. C

    Do we always need to exclude solutions that would give 0 in denominator?

    When working with fractions and when we have a fraction or equation with fractions like this one for example ##\frac{x}{x-1}+\frac{x}{x+1}=\frac{9}{4}## do we always need to set boundaries? Like, do we always need to write that x can't be a number that would give the denominator 0? In this...
  6. A

    B It works but why? (Matching experimental data to a random equation)

    Hey guys, I've about a week left to submit my final paper for my trade degree in transportation. The paper is about an analysis of potential implementation of an electric car for direct deliveries in my area where I live. In part of it, I try to analyze how many possible trips a car like...
  7. M

    Engineering Manipulating complex vectors

    Hi Here is my attempt at a solution for problems 1) and 2) that can be found within the summary. Problem 1) a = 3-2i b= -6-4i c= 4+ 6i d= -4+3i Now, to calculate each vector modulus, I applied the following formula: $$\left| Vector modulus \right| = \sqrt {(a^2 + b^2) }$$ where a = real part...
  8. pairofstrings

    I Object encapsulation

    Is it possible to encapsulate an object which has quantity, structure, space, change into a single equation? I am just asking if it is possible or not, I am not asking anyone to tell me how. Assume that there is a book. Is it possible to write everything about this book in a single equation...
  9. pairofstrings

    B LHS equal to RHS?

    Hello. My question is: what does it mean if an equation is being equaled to another equation? Please give me an example. Thanks. Thank you for pointing my mistake. I should have written "expression" instead of "equation". My question is: what does it mean if an expression is being equaled to...
  10. pairofstrings

    B Equations and Word-problems

    Hi. Can every word-problem be converted into equation? Can every equation be converted into word-problem? Thanks!
  11. pairofstrings

    B Complicated equation and Simple equation for the Same Curve?

    If I draw some arbitrary curve then that curve can be represented convolutedly in mathematical elements and it can also be represented in simple mathematical elements? Thanks!
  12. brotherbobby

    Which points on the plane satisfy the given equation?

    Given the equation : ##|y| x = x##. Two conditions are possible : (1) ##\underline{y\geq 0}## : ##xy = x\Rightarrow \boxed{y = 1}\; (x \neq 0)##. We note that except for zero, ##-\infty<x<+\infty## for this case. (2) ##\underline{y < 0}## : ##-xy = x\Rightarrow \boxed{y = -1}\; (x \neq 0)##...
  13. Y

    Engineering Equation for a DC motor driving an arm

  14. B

    Topological insulators and their optical properties

    I have tried to write down the boundary conditions in this case and looked into them. As conditions i) and ii) were trivial, i looked into iii) and iv) for information that I could use. But all I got was that for the transmitted wave to have an angle, the reflective wave should also have an...
  15. Boltzman Oscillation

    How can I create an equation in matlab for image processing?

    Here is the documentation for the 2DFFT: https://www.mathworks.com/help/matlab/ref/fft2.html how would I go about creating this formula on MATLAB to apply it on an image? My guess is that I need to create the equation and then multiply it to the image I need such as: $$U = VI$$ where V is my...
  16. S R Wilder

    Is it the Thévenin theorem?

    I just need the meaning of In.
  17. ArcHorizon

    Physics Modeling of a Gas

    This was the equation that they showed me. I thought P was for pressure, V for Volume, T for Temperature, R for Gas Constant, and n for the number of moles. Was I correct for the initials?
  18. Z

    Time needed for a pressured N2O cylinder to reach the apex of its travel as a projectile....

    Hello, I am trying to understand the maths/physics/chemistry behind this situation. Here is the scenario. I have 8 grams of pressurized N2O in a cylinder at 60 bar/ 900 psi. If the temperature stays constant (let's say 50-70°C, or at a temperature where the N2O can stay as pressurized as...
  19. N

    Python Convert an equation to Python

    I have several equations and need to convert it into Python. The problem is that I tried to plot a graph according to the equation. However, the graph that I get is not the same as the original one. In the paper, the equation of error probability for MIM attack is given by: First Image Second...
  20. komarxian

    Multivariable calculus problem

    Homework Statement Find the points on the surface xy^2z^3=2 that are closest to the origin Homework Equations The Attempt at a Solution x,y,z=/= 0, as when x,y,z = 0 it is untrue. Right?? Otherwise, I am very unsure as to how to approach this problem. Should I be taking partial derivatives...
  21. M

    How do I find a plane that contains two given lines?

    Homework Statement a. Find a point at where these lines intersect b. Find the equation of a plane that contains the two lines. Homework Equations r[/B] = <1,3,0> + t<3,-3,2> r = <4,0,2> + s<-3,3,0> The Attempt at a Solution I correctly found the point of intersection to be...
  22. José Ricardo

    Ellipse graphic

    Homework Statement Graph the ellipse 4x² + 2y² = 1 Homework Equations 4x² + 2y² = 1 The Attempt at a Solution 2x² + y²/2 = 1/2 I searched for exercises on Google, and i didn't find an equation like that. I watched videoleassons too but it didn't teach this type of equation.
  23. G

    Equation for the resolving power of a microscope?

    Hi I'm reading through a Quantum Mechanics textbook called Quantum Mechanics by Book by Alastair I. M. Rae and in the opening chapter it talks about the Heisenberg uncertainty principle and talks about how a measurement of position of a particle causes an uncertainty from the momentum due to the...
  24. L

    Range calculation for Airbus A320 aircraft

    Homework Statement Hi to all, the Task I struggle with, is the range calculation from an Airbus A320 with the Breguet Range Equation which is defined as: Homework Equations R = (cl/cd) * (V/(g*SFC)) * ln(w0/w1) with V = velocity g = gravity SFC = specific fuel consumption w0 =...
  25. concernedhuman

    B Is there such a thing as Gm/r?

    Since Fg = Gmm/r2and Coulomb's law being similar to that: Fe = kQq/r2, and we also have E = kQ/r2 and g = Gm/r2 being alike, I was wondering if there's anything that corresponds to the potential equation kQ/r. I converted it myself and figured that it's going to be Gm/r, and I'm not sure if a...
  26. hugo_faurand

    B Equation with modulus

    Hello everyone! I'd like to know if it's possible to solve an equation with a modulus like this one : $$ ( \frac{200}{15x}) mod 2 = 0 $$ Thanks in advance. Regards!
  27. Subrahmanyan

    To find the nature of roots of a quintic equation....

    The asks for us to find the nature of roots of the following equation ,i.e,rational or irrational nature of the roots: the Equation is : http://www5a.wolframalpha.com/Calculate/MSP/MSP7106108efdacf0gb84ie000039f29d33b6hie7ic?MSPStoreType=image/gif&s=56&w=63.&h=18.x^5+x=5 I have been able to...
  28. pairofstrings

    B What is the connection between x^2 and a square shape?

    Hello. The curve y = x2 is a parabola that looks like this: I have a shape Square that looks like this: What I am noticing is that if I consider the equation y = x2 and also the shape Square, I find that there is no connection between them but the equation y = x2 is pronounced as x-square...
  29. S

    Equation construct ( please)

    Hello everyone, I need some help (or guidance). I have an equation f(x) = A/(x2). I need to construct the equation g(f(x)) with following conditions: - when f(x) -> 0 then g(f(x)) ->1; - when f(x) -> Fmax then g(f(x))->0; (This is important: there is some fixed Fmax value at which g(f(x))...
  30. B

    Two pencils of planes have a common plane

    Homework Statement Find the value of the parameter α for which the pencil of planes through the straight line AB has a common plane with the pencil of planes through the straight line CD, where A(1, 2α, α), B(3, 2, 1), C(−α, 0, α) and D(−1, 3, −3). Homework Equations Let Δ be a line given by...
  31. shintashi

    B Equations vs. Functions Quadratic and Cubic?

    So if i take the rules that a straight vertical line drawn through the function with more than one intersection implies it is not a function, to mean that the quadratic equation for a circle is not a function. Furthermore, it also implies a cubic equation, such as x^3 can be a function, because...
  32. D

    B Showing/proving a physical relationship

    I derived a relationship between frequency and tension of a string, accounting for tension's effect in the linear density of the string. So in a nutshell, the equation is more complicated and is in the form of f^2=aT^2+bT (f is frequency, T is tension, ab are constants involving the control...
  33. 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...
  34. A

    I Empirical equation from two variables (1 input and 1 output)

    Hi, I have empirical data from my experiments. There are 2 columns of data (2 interdependent variables- temperature and viscosity).. 1 column (temperature) is input variable (temp. of tested material, once it was melted, it was gradually increased during the experiment). 1 column (viscosity)...
  35. SemM

    A How does one "design" a PDE from a physical phenomenon?

    Hi, I have read some on the PDEs for fluids, and particularly for rogue waves, where for instance the extended Dysthe equation and the NLSE look rather intimidating: Take for instance the Non-linear Schrödinger eqn: \begin{equation} \frac{\partial^2 u}{dx^2}-i\frac{\partial d...
  36. G

    Potential Energy Change

    Homework Statement A 95-kg mountain climber hikes up a mountain to an elevation of 5000 m. What is the change in the climber's potential energy? Homework Equations I might be missing something but here's everything that might be relevant: w=fx p=w/t p=f*v KE=1/2mv^2 PEg=mgh PEe=1/2kx^2 Wnet=E...
  37. G

    What is the relation of mass and power?

    How should power be calculated in a situation where distance and time are both given, as well as mass. At first, I was thinking of just using W=Fx (force as mass x 9.8) to solve for work, and then I would take the solution for work and put it into P=wt. Does this make sense? That's my best...
  38. S

    A Solving the Schrödinger equation for free electrons

    Dear all, sorry I made a new post similar to the previous post "Initial conditions..", however, a critical point was missed in the previous discussion: The initial conditions y(0)=1 and y'(0)=0 are fine and help in solving the Schrödinger equation, however, studying free electrons, the equation...
  39. S

    LaTeX Split up a horrible equation

    Hi, how do I split up this horrible equation into several lines? given in latex code: \Psi_2=...
  40. A

    Solution of an exponential equation

    Homework Statement I am trying to solve an equation. 128^b - 127^b = 147.058. Homework Equations The Attempt at a Solution I have tried numerical methods like Bisection method and Newton-Raphson, but I need analytical solution. Thank you.
  41. Alexander350

    B Solving a differential equation with a unit vector in it

    I need to solve: \dot{\mathbf{r}}=-kv\hat{r} - \dot{\mathbf{r}_s} However, I do not know how to deal with the fact that there is a unit vector. How can this be done? \dot{\mathbf{r}_s} is a constant vector.
  42. W

    Green's functions: Logic behind this step

    Homework Statement Hi all, I came across these steps in my notes, relating to a step whereby, $$\hat{G} (k, t - t') = \int_{-\infty}^{\infty} e^{-ik(x - x')}G(x-x' , t-t')dx$$ and performing the following operation on ##\hat{G}## gives the following expression, $$[\frac{\partial}{\partial t}...
  43. L

    Poisson's equation in 1D with point source

    Homework Statement Solve ##\Delta\phi = -q\delta(x)## on ##\mathbb{R}##. Correct answer: ##\phi = -\frac{q}{2}|x| + Ax + B## Homework Equations The Attempt at a Solution In one dimension the equation becomes ##\frac{d^2 \phi}{d x^2} = -q\delta(x)##. We integrate from ##-\infty## to ##x## to...
  44. CollinsArg

    B What does the Center of Mass equation mean?

    How can I understand logically this equations? it seems to me like an avarage of equation...but why each term is specifically in the way it is? Thank you
  45. W

    Differential Equations: LRC

    Homework Statement How does one show that q(t) is indeed a solution? Homework Equations The Attempt at a Solution My current idea is that i should come up with any form of solution, like q = Acos(ωt), and slot it in the RHS. Reason being that if q is indeed a solution, the result of the...
  46. kelvin490

    MATLAB Solving equation with integration using MATLAB

    I would like to solve an equation below using MATLAB: All the parameters except p are known, so I only need to solve for p. However since I need to consider the sign of the integrand and there is an absolute value sign in it I don't know how to solve it. Could anyone please help? Thank you.
  47. Erenjaeger

    I Proving equation for duration of transit

    The lecturer went through part of the proof so i am stuck on finishing it off. equation: ΔT = PR*/πα where P is the period of planet around star R* is radius of star α is orbital radius of planet what I have done so far... ω (orbital speed) = v/α also = 2π/p (p being a planetry year) and...
  48. pairofstrings

    B A single or multiple equations of 'y'?

    Hello. I have an object that I want to talk about. The object above does not seem to have numbers in a definite pattern, so, I think that this object will not have a single equation that describes the object, but instead there will be multiple number of equations of 'y' in 'x'. Correct?
  49. A

    Electric field of AC source

    Hello everybody I got really confused. I want to calculate the Electric-Field of an alternating current source( Sine(wt)). I do not know what is its equation. I need to run it in Matlab to get a result. Can you please give me some hints? I need to write the equation to calculate the E-field in...