What is equations: Definition and 37 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. billtodd

    The ratio between two horizontal speeds in this pendulum arrangement

    From the forces equation I can only understand from it that the forces' equations are:##N=Mg## and ##T\sin \theta=m\ell \ddot{\theta}##. But I don't know how to find the Torques' equations. Any help is appreciated. N=Mg ##Tcos⁡θ+N=mg
  2. G

    I Maxwell says that Div B = 0

    So if Maxwell's Equations state that Div B = 0 which is a statement that there are no magnetic monopoles why do some folks claim that Cosmic Inflation "dilutes" magnetic monopoles?
  3. chwala

    Work out the amount that Arjun paid in rent in 2019

    I am interested in a more direct linear equations approach...to solve, which i know is possible... A. My initial thinking was along the lines, let ##x## be the amount that Arjun paid in 2018 and let ##y## be the amount that Gretal paid in 2018 where A was the total amount paid in 2018 ...
  4. Safinaz

    Simpliying this partial differential equation

    Does this equation mean that: ## x+y =z ##, and ## x = y##? I mean ## \delta_{ij} ## terms in the LHS of the eqaution equal those at the RHS ? with knowing that ## \partial_i \partial_j x ## term dose not vanish for ## \delta_{ij}=1 ## Any help is appreciated!
  5. A

    Solve for all angles x: cos(2x) + cos(x) = 0, where 0<x<2pi

    I'm not sure how to go about solving this mathematically? In just using what seems obvious, I know the angle pi would work, because pi = -1, and 2pi = 1. However, as far as manipulating the equations in a way where it can solve itself without me having to look at a chart where cos for both x...
  6. T

    I Heisenberg Equations of Motion for Electron in EM-field

    Consider the Heisenberg picture Hamiltonian $$H(t) = \int_{\textbf{r}}\psi^{\dagger}(\textbf{r},t)\frac{(-i\hbar\nabla+e\textbf{A})^{2}}{2m}\psi(\textbf{r},t)$$ where ##\psi(\textbf{r},t)## is a fermion field operator. To find the equations of motion that ##\psi,\psi^{\dagger}## obey. I would...
  7. chwala

    I Solve the system of linear equations

    This is fine but i have my own approach as follows; ##\begin{pmatrix} -7 & -6 & -12 & -33 \\ 5 & 5 & 7& 24 \\ 1 & 0 & 4 & 5 \end{pmatrix} ## → Row 1 times 5 and row 2 times 7... ##\begin{pmatrix} -35 & -30 & -60 & -165 \\ 35 & 35 & 49& 168 \\ 1 & 0 & 4 & 5 \end{pmatrix}## →...
  8. paulb203

    Basic Simultaneous Equations Problem

    I got c=13, but I'm using an app that, at the 'diagnostic' stage, doesn't tell you if you're correct or not (I multiplied the first equation by 4, and the second by 2, then subtracted the second from the first)
  9. B

    Derivation for modified sine curve equations

    Hello, I am looking for a detailed derivation of the equations used to generate the modified sine curve. I found one in Cam design handbook by Harold A. Rothbart but I didn't understand how we get certain equations. My end goal is to combine the modified sine curve with constant velocity and get...
  10. Filip Larsen

    I Solving matrix commutator equations?

    I have the matrix relationship $$C = A^{-1} B^{-1} A B$$ I want to solve for ##A##, where ##A, B, C## are 4x4 homogeneous matrices, e.g. for ##A## the structure is $$A = \begin{pmatrix} R_A & \delta_A \\ 0 & 1 \end{pmatrix}, A^{-1} =\begin{pmatrix} R_A^\intercal & -R_A^\intercal\delta_A \\ 0 & 1...
  11. MatinSAR

    Find thermal capacity of a Van der Waals gas

    Can anyone guide me how can I find ##C_P## and ##C_V##? There are two equations in my book which I think can help but I'm not sure if I can use them to find ##C_P## and ##C_V## of a vander waals gas. I know equation of state of this gas.
  12. W

    Troubleshooting Circular Motion: Solving the Toy Car Loop Puzzle

    I tried using conservation of energy, and using the equations for circular motion, but I can't seem to find a solution. Any help?
  13. S

    B More examples of equations that unexpectedly model nature in similar ways

    Been dipping my toes into maths by examining how equations work on the most basic level, and I love encountering equations that turn out to model similar aspects in nature, for example the inverse square law is apparent in equations for gravity and for electromagnetism. In the thumbnail of...
  14. L

    B Which equation should be used to calculate impulse: I=F(t) or I=m1v1-m0v0?

    Just going over some basic equations to help someone out and i have noticed that i get 2 different answers when using 2 different but very similar equations for impulse. first eq. I=F*t lets say that i have a small rocket motor that has a force of 5.95 newtons and a impulse of 2.5 n-s...
  15. Ben2

    Show that the Poiseuille field of flow is rotational

    I tried using these equations, but it's not clear if we should hold y_1 = y_2. A transverse velocity vector would produce a flow at some angle to the horizontal, but How do they known there's such a vector?
  16. S

    A Exploring the Restricted Three Body Problem: Hamiltonian vs Lagrangian Approach

    I am having a difficult time finding papers that derive the equations of motions of the Restricted Three Body Problem using Newtonian mechanics. I have found some (barely) for the CRTBP, but there are none on the ERTBP. Does it depend on what is being studied within the problem that the...
  17. P

    A Quintic and Higher Degree Polynomial Equations

    What's the root formula for fifth and higher degree polynomial equations, which have roots in radicals?
  18. W

    I The electric field due to a dipole

    and Can someone explain how they made these equations like this? How did the radius become that equation? What formulas from algebra did they applied? I'm looking at these formulas and I don't understand how r=z+1/2*d
  19. E

    Why is the square root of x^2 = |x|?

    If I reason this as follows, I run into problems. Please help me understand what is wrong with reasoning like this. a) I start with the left hand side of the equation and let that x be -2. b) I square it. This gives me 4. So I now have the square root of 4. c) The square root of 4 is +/- 2. The...
  20. L

    Show that these equations are linearly dependent using Mathematica

    TL;DR Summary: solve linear systems of equations Hi, I am supposed to solve the following problem with Mathematica This is the tutorial we received on how to solve linear systems with Mathematica: I then tried the whole thing for the task, but unfortunately Mathematica doesn't give me any...
  21. M

    B Solving Simultaneous equations with cos() and sin()

    Hello I have two equations given below and answer is also given solve by simultaneously i am not sure how this happen please guide thank you
  22. brotherbobby

    LaTeX 1. Colour while using ##\LaTeX## 2. Preview first time

    I have observed that one cannot use colour here while typing something in ##\LaTeX##. Let me try : ##\text{Let's see if this can be \color{blue} {coloured}?}##. Nope, I type, via an edit after posting. Of course I can use colour from the panel above, but that will have to work for the entire...
  23. barryj

    I Using an array to solve a system of equations

    I was given a problem to use an array, not a matrix, to solve a system of equations. I have not heard of doing this and can not find any reference to how to do it. like... 2x+3y = 12, and 3x-4y = 14 How is this solved using an array or grid?
  24. K

    Finding value of g given a motion map

    I tried using all equations of motion but couldn't get the correct answer. Any hint would greatly help. I've tried doing this for two days now!
  25. J

    B Understanding the Relationship Between i*cos and sin in Circuit Analysis

    In circuit analysis, everything seems to work out when you set i*cos = sin. But thats not a legitimate equation, so why does that work? Is there a proof that this is a real equation?
  26. D

    I Dimension and solution to matrix-vector product

    Let $$ X \in R^{m*n} $$ where m=n with rank(X)<m then there is at-least one equation which can be written as a linear combination of other equations. Let $$ \beta \in R^{n} $$. $$ X\beta=y $$ Suppose we have x<m independent equations (the equations are consistent) formed by taking the dot...
  27. Connor123

    A Level: Rearranging Equations into Y = mx + c

    c = fλ, where the dependent (y) variable is λ, and the independent variable (x) is f. λ = c/f c * 1/f = λ Are you able to leave 1/f as equavalent to simply 'x', where the gradient would be c, or do you have to ensure 'x'/independent variable is actually f?
  28. SwetS

    A Addition of two velocities

    I am going through https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf, here I don't understand how we wrote eq. 1.5.3 with the help of three-vectors to get eq. 1.5.4.
  29. J

    B Question about Euler's formula

    Euler gave us the below equations: But this doesn't actually give me a number value for where the y value is when you plug in a number for x. For example, if i plug in 2pi for x, i know cosx should be 1. But that equation gives me (e^2i*pi +e^-2i*pi)/2. This doesn't give me 1. So what really...
  30. D

    What is the Resultant Force from adding these two forces?

    when is apply the formulae of resultant force for the two equations the angles cancel out and I can't get any value for the angle when i add the two equations: R^=f1^2+f2^2+2f1f2cos(theta) squart3*R=f1^2+f2^2-2f1f2cos(theta)
  31. pairofstrings

    I Questions about these Trigonometry Graphs involving sin() and cos()

    Hi. I have two trigonometric equations whose graphs I am trying to understand. Here are the equations: 1. a sin(x) - b cos(y) = y; a = 2, b = 2 2. a sin(x) + b cos(y) = 1; a = 1, b = 1 My question is why the graphs are the way they are. What should I do to understand them? Can anyone...
  32. JD23

    A Stimulated emission/absorption - which equations act on external targets?

    Stimulated emission-absorption have two symmetric equations for population (N_1 - number of ground state atoms, N_2 - of excited) written below, at heart of lasers: https://en.wikipedia.org/wiki/Stimulated_emission#Mathematical_model We assume that both equations act on the central target inside...
  33. chwala

    Find the area of the quadrilateral OCBAO

    My challenge was on trying to determine the angles: My approach; came up with a number of equations: ie ##m+n=70^0## ##r=p+40^0## ##q-2r=100^0, ⇒ r=50^0 + \dfrac{1}{2} q## then it follows that, ##2q+100^0=180^0## ##⇒q=40^0, r=70^0, p=m=30^0, n=40^0## ##m+40^0+t=180^0, ⇒t=110^0## and...
  34. Slimy0233

    [Kinematics] Calculating the maximum height reached by the ball

    I realize I can solve the other way too. But I want to solve using the equations ##v = u +at## ##S = ut +0.5(at^2)## and I don't know why I didn't get the right answer. Thank you for your help!
  35. Edy56

    Engineering Confused about defining unique current loops when using the node voltage method

    Please only respond if you know the node voltage method. I need to write equations for this circuit. My problem is that 1 has two paths towards 3, so when I write equations do I write both of those paths or only one. U30(1(-jXc1)+1/(jXl1+R1)+1/(jXl2)-U10((1/(-jXc1)+1/(R1+jXl1))-U20.... (the...
  36. Edy56

    Engineering Power dissipation of 2 diodes (1 is a Zener diode)

    Are my equations correct? And my process of how I should solved this problem?
  37. C

    Direct proportionality equations

    For this, Can a the equation ##x = vt + x_0## not be considered a direct proportionality between ##x## and ##v##? If so, is it because there is a constant ##x_0##? Many thanks!
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