Equation is called the mathematical equality that exists between two expressions, this is made up of different elements both known (data) and unknown (unknowns), which are related through mathematical numerical operations. The data is generally represented by coefficients, variables, numbers and constants, while the unknowns are indicated by letters and represent the value that you want to decipher through the equation. Equations are widely used, mainly to show the most exact forms of mathematical or physical laws, which express variables.
What is equation
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The term comes from the Latin "aequatio", whose meaning refers to equalize. This exercise is a mathematical equality existing between two expressions, these are known as members but they are separated by a sign (=), in these, there are known elements and some data or unknowns that are related through mathematical operations. Values are numbers, constants, or coefficients, although they can also be objects such as vectors or variables.
The elements or unknowns are established through other equations, but with an equation solving procedure. A system of equations is studied and solved by different methods, in fact, the same happens with the equation of the circumference.
History of equations
The Egyptian civilization was one of the first to use mathematical data, because by the 16th century they already applied this system, to solve problems associated with the distribution of food, although they were not called equations, it could be said that it is the equivalent to the current time.
The Chinese also had knowledge of such mathematical solutions, because at the beginning of the era they wrote a book where various methods were proposed for solving second and first grade exercises.
During the Middle Ages, the mathematical unknowns had a great boost, since they were used as public challenges among the expert mathematicians of the time. In the 16th century, two important mathematicians made the discovery of using imaginary numbers to solve the second, third and fourth degree data.
Also in that century Rene Descartes made the scientific notation famous, in addition to this, in this historical stage one of the most popular theorems in mathematics was also made public "Fermat's last theorem".
During the seventeenth century the scientists Gottfried Leibniz and Isaac Newton made possible the solution of the differential unknowns, which gave rise to a series of discoveries that occurred during that time regarding these specific equations.
Many were the efforts that mathematicians made until the beginning of the 19th century to find the solution to the equations of the fifth degree, but all were failed attempts, until Niels Henrik Abel discovered that there is no general formula to calculate the fifth degree, also during this time physics used differential data in integral and derived unknowns, which gave rise to mathematical physics.
In the 20th century, the first differential equations with complex functions used in quantum mechanics were formulated, which have a wide field of study in economic theory.
Reference should also be made to the Dirac equation, which is part of the studies of relativistic waves in quantum mechanics and was formulated in 1928 by Paul Dirac. The Dirac equation is fully consistent with the special theory of relativity.
Equation characteristics
These exercises also have a series of specific characteristics or elements, among them, the members, terms, unknowns and solutions. The members are those expressions that are right next to the equals signs. The terms are those addends that are part of the members, likewise, the unknowns refers to the letters and finally, the solutions, which refer to the values that verify equality.
Types of equations
There are different types of mathematical exercises that have been taught at different levels of education, for example, the equation of the line, chemical equation, balancing of equations or the different systems of equations, however, it is important to mention that these are classified into algebraic data, which in turn can be of first, second and third degree, diophantine and rational.
Algebraic equations
It is a valuation that is expressed in the form of P (x) = 0 in which P (x) is a polynomial that is not null but not constant and that has integer coefficients with a degree n ≥ 2.
- Linear: it is an equality that has one or more variables in the first power and does not need products between these variables.
- Quadratic: it has an expression of ax² + bx + c = 0 having a ≠ 0. here the variable is x, ya, b and c are constants, the quadratic coefficient is a, which is different from 0. The linear coefficient is b and the term independent is c.
It is characterized by being a polynomial that is interpreted through the equation of the parabola.
- Cubic: cubic data that have an unknown are reflected in third degree with a, b, c and d (a ≠ 0), whose numbers are part of a body of real or complex numbers, however, they also refer to rational digits.
- Biquadratic: It is a single variable, fourth degree algebraic expression that has only three terms: one of degree 4, one of degree 2 and an independent term. An example of a biquad exercise is the following: 3x ^ 4 - 5x ^ 2 + 1 = 0.
It receives this name because it tries to express what will be the key concept to delineate a resolution strategy: bi-square means: "twice quadratic." If you think about it, the term x4 can be expressed as (x 2) raised to 2, which gives us x4. In other words, imagine that the leading term of the unknown is 3 × 4. Similarly, it is correct to say that this term can also be written as 3 (x2) 2.
- Diophantines: it is an algebraic exercise that has two or more unknowns, in addition, its coefficients encompass all the integers of which the natural or integer solutions must be sought. This makes them part of the entire number group.
These exercises are presented as ax + by = c with the property of a sufficient and necessary condition so that ax + by = c with a, b, c belonging to the integers, have a solution.
- Rational: they are defined as the quotient of the polynomials, the same ones in which the denominator has at least 1 degree. Speaking specifically, there must be even one variable in the denominator. The general form that represents a rational function is:
In which p (x) and q (x) are polynomials and q (x) ≠ 0.
- Equivalents: it is an exercise with mathematical equality between two mathematical expressions, called members, in which known elements or data appear, and unknown elements or unknowns, related by mathematical operations. The values of the equation must be made up of numbers, coefficients, or constants; like variables or complex objects such as vectors or functions, new elements must be constituted by other equations of a system or some other procedure for solving functions.
Transcendent equations
It is nothing more than an equality between two mathematical expressions that have one or more unknowns that are related through mathematical operations, which are exclusively algebraic and have a solution that cannot be given using the specific or proper tools of algebra. An exercise H (x) = j (x) is called transcendent when one of the functions H (x) or j (x) is not algebraic.
Differential equations
In them the functions are related to each of their derivatives. The functions tend to represent certain physical quantities, on the other hand, the derivatives represent rates of change, while the equation defines the relationship between them. The latter are very important in many other disciplines, including chemistry, biology, physics, engineering and economics.
Integral equations
The unknown in the functions of this data appear directly in the integral part. The integral and differential exercises have a lot of relationship, even some mathematical problems can be formulated with either of these two, an example of this is the Maxwell viscoelasticity model.
Functional equations
It is expressed by combining unknown functions and independent variables, in addition, both its value and its expression have to be solved.
State equations
These are constitutive exercises for hydrostatic systems that describe the general state of aggregation or increase of matter, in addition, it represents a relationship between the volume, temperature, density, pressure, state functions and the internal energy associated with matter..
Equations of motion
It is that mathematical statement that explains the temporal development of a variable or group of variables that determine the physical state of the system, with other physical dimensions that promote the change of the system. This equation within the dynamics of the material point, defines the future position of an object based on other variables, such as its mass, speed or any other that may affect its movement.
The first example of an equation of motion within physics was using Newton's second law for physical systems made up of particles and point materials.
Constitutive equations
It is nothing more than a relationship between the mechanical or thermodynamic variables existing in a physical system, that is, where there is tension, pressure, deformation, volume, temperature, entropy, density, etc. All substances have a very specific constitutive mathematical relationship, which is based on internal molecular organization.
Solving equations
To solve the equations, it is completely necessary to find their solution domain, that is, the set or group of values of unknowns in which their equality is fulfilled. The use of an equation calculator can be used because these problems are usually expressed in one or more exercises.
It is also important to mention that not all these exercises have a solution, since it is quite likely that there is no value in the unknown that verifies the equality that has been obtained. In this type of case, the solutions of the exercises are empty and it is expressed as an unsolvable equation.
Examples of equations
- Movement: at what speed must a racing car travel to travel 50km in a quarter of an hour? Since the distance is being expressed in kilometers, the time must be written in units of hours to have the speed in km / h. Having that clear, then the time that the movement lasts is:
The distance the car travels is:
This means that its speed must be:
The formula is:
Therefore, we must leave the "n", and we obtain:
Then the data is substituted:
And the amount of number of moles is 13.64 moles.
Now the mass must be calculated. As it is hydrogen gas, reference must be made to its atomic weight or molar mass, which is a diatomic molecule, made up of two hydrogen atoms.
Its molecular weight is 2 g / mol (due to its diatomic characteristic), then it is obtained:
That is, a mass of 27.28 grams has been obtained.
- Constitutive: there are 3 bars attached to a rigid beam. The data are: P = 15,000 lbf, a = 5ft, b = 5ft, c = 8ft (1ft = 12 inches).
The solution is that it is assumed that there are small deformations and that the screw is totally rigid, that is why when applying the force P the beam AB will rotate rigidly according to point B.
