Summary
In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign. When seeking a solution, one or more variables are designated as unknowns. A solution is an assignment of values to the unknown variables that makes the equality in the equation true. In other words, a solution is a value or a collection of values (one for each unknown) such that, when substituted for the unknowns, the equation becomes an equality. A solution of an equation is often called a root of the equation, particularly but not only for polynomial equations. The set of all solutions of an equation is its solution set. An equation may be solved either numerically or symbolically. Solving an equation numerically means that only numbers are admitted as solutions. Solving an equation symbolically means that expressions can be used for representing the solutions. For example, the equation x + y = 2x – 1 is solved for the unknown x by the expression x = y + 1, because substituting y + 1 for x in the equation results in (y + 1) + y = 2(y + 1) – 1, a true statement. It is also possible to take the variable y to be the unknown, and then the equation is solved by y = x – 1. Or x and y can both be treated as unknowns, and then there are many solutions to the equation; a symbolic solution is (x, y) = (a + 1, a), where the variable a may take any value. Instantiating a symbolic solution with specific numbers gives a numerical solution; for example, a = 0 gives (x, y) = (1, 0) (that is, x = 1, y = 0), and a = 1 gives (x, y) = (2, 1). The distinction between known variables and unknown variables is generally made in the statement of the problem, by phrases such as "an equation in x and y", or "solve for x and y", which indicate the unknowns, here x and y. However, it is common to reserve x, y, z, ... to denote the unknowns, and to use a, b, c, ... to denote the known variables, which are often called parameters.
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Ontological neighbourhood
Related courses (31)
MATH-111(e): Linear Algebra
L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications.
MATH-250: Numerical analysis
Construction et analyse de méthodes numériques pour la solution de problèmes d'approximation, d'algèbre linéaire et d'analyse
EE-548: Audio engineering
This lecture is oriented towards the study of audio engineering, with a special focus on room acoustics applications. The learning outcomes will be the techniques for microphones and loudspeaker desig
Show more
Related lectures (180)
Nonlinear Equations: Fixed Point Method
Covers the topic of nonlinear equations and the fixed point method.
Block Pulled by a Spring: Dynamics
Covers the dynamics of a block connected to a spring, deriving equations of motion and solving for key time points.
Acoustic Simulation: Pulsating Sphere
Covers the simulation of acoustic waves in fluids using the Pressure Acoustics, Frequency Domain interface in COMSOL Multiphysics.
Show more
Related publications (51)

FENNECS: a novel particle-in-cell code for simulating the formation of magnetized non-neutral plasmas trapped by electrodes of complex geometries

Stefano Alberti, Jean-Philippe Hogge, Joaquim Loizu Cisquella, Jérémy Genoud, Francesco Romano

This paper presents the new 2D electrostatic particle-in-cell code FENNECS de- veloped to study the formation of magnetized non-neutral plasmas in geometries with azimuthal symmetry. This code has been developed in the domain of gy- rotron electron gun des ...
2024

High-frequency response of a hemispherical grounding electrode

Marcos Rubinstein, Antonio Sunjerga

The lightning discharge current is characterized by a high-frequency spectrum extending from DC to about 10 MHz. The calculation of the grounding impedance is one of the most important aspects of designing lightning protection systems. The search for analy ...
2024

Testing Benchmark for a Block Conjugate Gradient Solving Scheme in Poroelasticity

With the advancement in fields of science more complex and more coupled phenomena can be explained, calculated and predicted. To solve these problems one has to update the related tools. Since analytical solutions do not exist for all physical processes, s ...
2023
Show more
Related concepts (16)
Algebra
Algebra () is the study of variables and the rules for manipulating these variables in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields.
Factorization
In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial factorization of x2 – 4. Factorization is not usually considered meaningful within number systems possessing division, such as the real or complex numbers, since any can be trivially written as whenever is not zero.
Root-finding algorithms
In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function f, from the real numbers to real numbers or from the complex numbers to the complex numbers, is a number x such that f(x) = 0. As, generally, the zeros of a function cannot be computed exactly nor expressed in closed form, root-finding algorithms provide approximations to zeros, expressed either as floating-point numbers or as small isolating intervals, or disks for complex roots (an interval or disk output being equivalent to an approximate output together with an error bound).
Show more
Related MOOCs (7)
Trigonometric Functions, Logarithms and Exponentials
Ce cours donne les connaissances fondamentales liées aux fonctions trigonométriques, logarithmiques et exponentielles. La présentation des concepts et des propositions est soutenue par une grande gamm
Trigonometric Functions, Logarithms and Exponentials
Ce cours donne les connaissances fondamentales liées aux fonctions trigonométriques, logarithmiques et exponentielles. La présentation des concepts et des propositions est soutenue par une grande gamm
Plasma Physics: Introduction
Learn the basics of plasma, one of the fundamental states of matter, and the different types of models used to describe it, including fluid and kinetic.
Show more