# What is equation: Definition and 84 Discussions

In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with 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 well-formed formula consisting of two expressions related with an equals sign 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.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. ### Vector parametric equation of line

I can imagine x + y = 1 to be line in xy - plane but how can x + 2y + z = 3 be a line, not a plane? Thanks
2. ### A Teleparallel Gravity

Where can I find detailed derivation of Field Equation of Teleparallel Gravity from variation of Action ?
3. ### B Implication vs Equivalence

I've seen a lot of people use implication and equivalence logic incorrectly. For example, when solving equations (i.e. ##x - 2 = 3 \implies x = 5##). Implication is not reversible, thus it only works in one way. By saying, ##x - 2 = 3 \implies x = 5##, you are essentially saying that it is...
4. ### How to balance an equation for the incomplete combustion of acetic acid?

Balancing the complete combustion of acetic acid equation to carbon dioxide and water is straightforward if you remember hydrogen is always +1, elemental oxygen is zero and combined oxygen is always -2. just balance the exchange of electrons between oxygen and carbon. But how do you balance...
5. ### I What year were Navier-Stokes equations introduced?

Who and when first time introduced below equations(dont have to be in same notation, content is important)? If this formula is always the same, what is contribution of Navier, what of Stokes, what changes all these years?
6. M

### Euler- Lagrange equation proof

For this problem, The solution is, However, I have a question about the solution. Does someone please know why they write out ##\frac{dF}{dx} = \frac{\partial F}{\partial y}y' + \frac{\partial F}{\partial y'}y''## since we already know that ##\frac{dF}{dx} = 0##? Thanks!
7. ### The value of (b - c) / (c - a)

$$(b-a)^2-4(b-c)(c-a)=0$$ $$b^2-2ab+a^2=4(bc-ab-c^2+ac)$$ $$b^2-2ab+a^2+4ab=4bc-4c^2+4ac$$ $$(b+a)^2-4ac=4c(b-c)$$ $$b-c=\frac{(b+a)^2-4ac}{4c}$$ I don't know how to continue and not even sure what I did is useful. Thanks
8. ### I Separation of variables and the chain rule

Hi; given the equation ydy/dx=x^2 how is the chain rule applied to result in ydy =x^2dx? Thanks
9. ### I Help understanding Maxwell's Equations please

I have having trouble understanding Maxwell's Equations. Can anyone recommend some good book or website that can help me to understand these Equations? How can electric and magnetic fields travel perpendicular to each other? What causes electromagnetic waves to first radiate from its source? I...
10. ### Two Trains and a Bee: Distance Question

This is a question from the MIT Open courseware website. (1). d = vt + ut let t = time it takes d = (u + v)t t = d / (u + v) (2). d = vt + ut d - vt = ut. Substitute t with d / (u + v) d - v*(d/(u+v)) = u*(d/(u+v)) d - v*(d/(u+v)) = “distance...
11. ### Bees and Trains: A distance problem

This question is from the MIT Courseware. I’m having difficulty finding the general equation to solve the problem (1). d = vt + ut d = (u + v)t t = d/(u + v) (2). d = vt + ut d - vt = ut sub t with d/(u+v) d - (v*d)/(u+v) = (u*d)/(u+v) I’m done with the...
12. ### Solve the given trigonometry equation

I was able to solve with a rather longer way; there could be a more straightforward approach; My steps are along these lines; ##\sinh^{-1} x = 2 \ln (2+ \sqrt{3})## ##\sinh^{-1} x = \ln (7+ 4\sqrt{3})## ##x = \sinh[ \ln (7+ 4\sqrt{3})]## ##x = \dfrac {e^{\ln (7+ 4 \sqrt{3})} - e^{-[\ln 7+ 4...
13. ### Mathematica Solving a complicated equation for approximate analytical Solution using Mathematica

Hello there, I am trying to solve the Following equation for r, $$2 a Q^4+5 r^4 \left(3 c (\omega +1) r^{1-3 \omega }-2 r (r-3 M)-4 Q^2\right)=0$$ Clearly this is unsolvable. But if we substitute a=0 and c=0 we get one of the solution, ##r=\frac{1}{2} \left(\sqrt{9 M^2-8 Q^2}+3 M\right)##. Can I...
14. ### Simultaneous equation involving cos, sin

Thanks a lot in advance!
15. ### Solve the given first order PDE

Solve the given PDE for ##u(x,t)##; ##\dfrac{∂u}{∂t} +10 \dfrac{∂u}{∂x} + 9u = 0## ##u(x,0)= e^{-x}## ##-∞ <x<∞ , t>0## In my lines i have, ##x_t = 10## ##x(t) = 10t+a## ##a = x(t) - 10t## also, ##u(x(t),t)= u(x(0),0)e^{-9t}## note this is from, integrating ##u_t[u(x(t),t] =...
16. ### Solve the given first order Partial differential equation.

Solve the given PDE for ##u(x,t)##; ##\dfrac{∂u}{∂t} +8 \dfrac{∂u}{∂x} = 0## ##u(x,0)= \sin x## ##-∞ <x<∞ , t>0## In my working (using the method of characteristics) i have, ##x_t =8## ##x(t) = 8t + a## ##a = x(t) - 8t## being the first characteristic. For the second...
17. ### Confusion about variables in polar coordinates

My confusion refers to this question above. If I were to ask you, what is the equation of the radial line, what would you say? I know that the general equation the radial line with cartesian gradient of m has an equation of θ = arctan(m). Clearly here the angle between the radial line and...
18. ### How do I use LaTex in the forum for to write equations?

Hello. I registered today. Maybe you can help me. How do I use LaTex in the forum for to write equations?
19. ### Help with Recurrence Equation

Hi there, I am going through a book on multi-storey steel structures and I have come to a chapter that gives approximate methods to calculate rotations at the joints (The intersecting members) of a rigid frame. There is a recurrence equation that computes the rotations and this is given below...
20. ### I Normal Mode calculation steps

Can someone please explain me the steps of calculation of X1:X2 after putting in the lower value of W^2 in equation 9.9 in "Riley, Hobson, Bence - Mathematical Methods for Physics and Engineering 2006 - pg 319"? I have attached the page as a PDF file. Thank you.

22. M

### Solving linear DE systems using fundamental matrix

For this problem, I am confused by the term below. I get all their terms, expect replacing the highlighted term by ##e^{3t}##, does someone please know whether this is yet another typo? Thanks!

24. ### Is this a simultaneous equation question?

hb=54 2h+2b=33 h=54/b therefore, 2(54/b)+b=33 108/b + b = 33 I’ve got a feeling I’ve gone down a blind alley here. Any hints?
25. ### Find the equation of the tangent plane and normal to a surface

In my line i have, ##\dfrac{∂r}{du} = \vec{i} +\dfrac{1}{2}u \vec{k} = \vec{i} +1.5 \vec{k}## ##\dfrac{∂r}{dv} = \vec{j} -\dfrac{1}{2}v \vec{k} = \vec{j} -0.5\vec{k}## The normal to plane is given by, ##\dfrac{∂r}{du}× \dfrac{∂r}{dv} = -\dfrac{3}{2} \vec{ i} + \dfrac{1}{2}\vec{j}+\vec{k}##...
26. ### Looking for equation/formula for calculating boomerang trajectory based on few variables for game

So basically i am vary green when it comes to equations/formula Well to the point i don't even know if proper name for what i'm looking for is called equation or formula but let's stick to equation So going as simple as i can I want to throw boomerang creating boomerang like ellipse trajectory...
27. ### Electric field vector equation: Finding the neutral point for two charges

This is the general suggested approach given in a textbook. My question is why can I not directly write it in vector form? E1 vector + E2 vector =0 should be valid no? Why are they choosing to write E1 mag + E2 mag=0 Then find a vector form Then convert the magnitude equation into a vector...
28. M

### Lagrange equation for block and incline

For this problem, Does someone please know where the term highlighted in blue came from? Thanks!
29. ### Chemistry Need help with Ideal Gas Question

Need help solving this question. Can't seem to get the right answer using PV/T=constant P1V1/T1 = P2V2/T2 Patm = 75.23cmHg T1+20+273=293K STP: P=1.01 x 10^5 N/m^2 Pabs=41cmOil P1 = density x g x h = (810 kg/m^3)(9.8 m/s^2)(75.23-41)x10^-2 mOil=2717.18 N/m^2...
30. ### What is the energy equation in Schrodinger's Spherical equation?

I attempted the problem by first finding the radial, theta, and phi equation for the ground state of a hydrogen atom. I multiplied the three equations to get the wave equation. From there, I took each derivative in the Schrodinger Spherical equation and found that ## \frac {\partial^2 \psi}...
31. ### I Runge-Kutta 4 w/ some sugar on the top: How to do error approximation?

Hello! I'm currently working with a problem which allows modelling ball motion \begin{aligned} m \ddot{x} & =-k_x \dot{x} \sqrt{\dot{x}^2+\dot{y}^2} \\ m \ddot{y} & =-k_y \dot{y} \sqrt{\dot{x}^2+\dot{y}^2}-m g \end{aligned} Given that ##k_x, k_y=0.005##, ##m=0.01## and ##g=9.81## and when...
32. ### Find the solution to the given differential equation

I need insight on the highlighted in Red on how ##\left[\dfrac{dz}{dx} - 1 = \dfrac{dy}{dx}\right]## otherwise the rest of the steps are clear. I just read that ##\dfrac{dx}{dy} \dfrac{dy}{dz} \dfrac{dz}{dx} =-1##
33. ### Show proof of point C in the given problem that involves Polar equation

c Parts (a) and (b) are okay ... though the challenge was on part (a) My graph had a plot of r on the y-axis vs θ on the x-axis). The sketch of my graph looks like is shown below; I suspect the ms had θ on the x-axis vs r on the y-axis. I used the equation ##r=\sqrt{\dfrac {1}{θ^2+1}}##...
34. ### Find the equation of the invariant line through the origin

My approach - i think similar to ms approach. The required Equation will be in the form ##y=mx## ##\begin{pmatrix} a & b^2 \\ c^2 & a \end{pmatrix} ⋅ \begin{pmatrix} k \\ mk \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} ## ##ak+b^2mk=x## ##kc^2+amk=y## ##x=k(a+b^2m)##...
35. M

### Reduction of order for Second Order Differential Equation

For this, I tried solving the differential equation using an alternative method. My alternative method starts at ##tv^{''} + v^{'} = 0## I substitute ##v(t) = e^{rt}## into the equation getting, ##tr^2e^{rt} + re^{rt} = 0## ##e^{rt}[tr^2 + r] = 0## ##e^{rt} = 0## or ##tr^2 + r = 0## Note that...
36. ### Solve the problem involving the velocity - time graph

For part (a) I came up with a simultaneous equation, i.e ##m+x+4m+700## ##5m+x=700## and ##15000=\dfrac{1}{2}[5m+2x]25## ##1200=5m+2x## therefore on solving the simultaneous, ##5m+x=700## ##1200=5m+2x## we get ##x=500## and ##m=40## the ms approach is here; more less similar...

45. ### B What went wrong with (-x)^2=x^2?

I have a very basic confusion that supports some basic elements of algebra. Being a high school student my teacher couldn't answer this, hope someone could help here. We know this equation is true: (-x)^2=x^2 but once we square root both sides it becomes this: -x=x we can see this equation was...
46. ### B Parametric Representation of Lines

I am having trouble with the concept that the equation L = {x + tv} is the more general form of the more familiar y = mx + b (In the first equation there should be a vector sign above the x and the v). It's hard for me to see the similarities between these two equations. 1: Even if we are...
47. ### The Von Mises stress equation on wikipedia does not balance out

On Wikipedia for Von Mises stress, it shows the following equation: But this does not work out. If I expand the second term I get:  \sigma_v^2 =...
48. ### Solve the given problem involving logarithms

In my working i have, ##\dfrac{\log_{11} x }{\log_{11} 4}= \log_{11} (x+6)## ##\dfrac{\log_{11} x }{0.5781}= \log_{11} (x+6)## ##\log_{11} x = \log_{11} \left[(x+6)\right]^{0.5781}## ##x^{1.729} = x+ 6## ##x^{1.729} -x-6=0## Having ##f(x) = x^{1.729} -x-6## At this point i made use of...
49. ### Homework help: Dropping a sand bag from a Hot Air Balloon

In this situation should my free fall equation contain the v0 of the baloon or I should deny it. Because it seems to me that there is no outer force acts on the sandbag, so the scenario is just the same as I climb to the same height at time t=0 and drop the sandbag at rest.
50. ### I What does a lower case r mean over variables in an equation?

I've attached an example here.