1. Statement that two mathematical expressions are equal (1+1 = 2). A problem is considered when one or more values are unknown and it is necessary to carry out a series of mathematical operations looking for the solution of the unknown.
2. Equalization between two things.
Etymology: By the Latin forms aequatio, aequationisregarding the verb aequarefor ‘balance’, ‘equalize’, on the adjective given as aequuswhich refers to ‘equal’, ‘fair’.
Grammatical category: noun fem.
in syllables: equation.
Equation
Evelyn Maitee Marin
Industrial Engineer, MSc in Physics, and EdD
Equations are mathematical expressions that describe the behavior of a variable or the relationship between two or more variables. The equations can vary in their complexity and application, and can be as simple as a linear equation, or move to more complex levels as a nonlinear differential integro equation.
Equations have accompanied humanity since ancient times, since there is evidence that many primitive civilizations already applied simple equations for their designs and daily tasks. At present, practically all academic training requires knowledge of equations and their clearances.
elements of an equation
Not all mathematical expressions are equations, since these expressions must contain a series of basic elements to be considered equations. These elements are:
• Equality: Equations is not just about combining numbers or variables. It is essential that they contain an equality that relates the members of both sides of it.
• Members: the equalities of an equation must always have elements on both sides. What is on each side of the equality are called members.
• Terms: refers to the elements of an equation separated by an addition or subtraction sign. They can be constants, variables or the product (or quotient) of a constant by a variable.
• Variable: every equation must contain at least one variable, sometimes also called unknowns.
• Constants: these are the values represented by numbers or letters that can appear alone in some term or as coefficients of a variable. If a variable appears alone in a term of an equation, it is understood that it has the number 1 as a coefficient.
The image shows a simple linear equation showing its fundamental elements: members, equality, terms, constants, and variables.
types of equations
The diversity of equations that are used in the world of sciences are of a very varied nature and complexity. Some equations do not have a specific interpretation in the physical world, but many others have been developed to try to represent a situation, phenomenon or physical model. For this reason, many areas such as Mathematics, Physics, Chemistry, medicine, statistics, geography, Biology, architecture, even art, can be manifested through equations or rely on them for their development. Here are some types of equations:
Types of equations according to their coefficients
One criterion to classify the equations is considering the coefficients that make it up, since as mentioned, the constants can be represented by numbers or letters; that is, the letters in an equation do not necessarily represent variables or unknowns. Variables are usually designated by the last letters of the alphabet and constants by the first. From this point of view, the equations can be literal or numerical:
Literal equations: are those that contain at least one coefficient denoted with a letter, for example, the equation to determine the kinetic energy (Ec) of a body is:
\({E_c} = \frac{1}{2}m{v^2}\)
where m is the mass and v is the velocity.
Depending on the case, the mass can be considered as a constant and the variable is the speed.
One of the most famous equations in modern physics is the energy equation developed by Albert Einstein, which establishes the equivalence between mass, the energy of a body, and the speed of light (c).
Numerical equations: In this type of equations, all the constants and coefficients of the terms are numerical values and the letters are only used for the variables. For example, the equation to determine the volume (V) of a sphere:
\(V = \frac{4}{3}\pi {r^3}\)
Where the variable is r, which represents the radius of the sphere, and although a symbol appears to denote the constant (pi), this is actually a numerical value, only that it has the particularity of being an irrational number (with infinite decimal places). ) and it is simpler to represent it with the Greek letter (pi).
Types of equations according to the functions they contain
Polynomial equations: this type of equations is characterized by the presence of polynomials in its terms, that is, the terms are made up of powers that contain coefficients, bases, and exponents. The exponent of greatest value determines the degree of the polynomial. One of the simplest polynomial equations is that of the line (y = mx + b).
Example of polynomial equation: \(8{x^6} – \frac{1}{5}{x^3} = 9{x^7} + 21\)
Equations with radicals: these equations owe their name to the fact that they have the variable inside a sign of a root. For its analysis and development, it is important to consider the properties of the roots and powers, considering that the roots are fractional exponents.
Example of equations with radicals: \(4 – \sqrt {5{x^6} – 13} = 17 + 11{x^5}\)
Trigonometric Equations: As the name implies, these equations are made up of trigonometric functions. They are widely used in the field of electrical engineering and physics to describe wave phenomena.
Example of trigonometric equations: \(y = 36se{c^2}\theta – 14tan\theta \)
Differential equations: this type of equations contains the derivatives of a function, as well as the function itself or its variables. These equations are very useful to describe transitory phenomena or that reflect the variation of one variable with respect to another.
Example of differential equations: \({e^x}\frac{{{d^2}y}}{{dx}} – 2{\left( {\frac{{dy}}{{dx}}} \ right)^4} + xy = 0\)
Integral Equations: Equations of this type contain functions within an integral operator, as well as the specification of the corresponding limits of integration.
Example integral equation: \(M = \mathop \smallint \nolimits_{{x_1}}^{{x_2}} Vdx\)
Sometimes, it is common to find equations that result from the combination of two or more of the described functions, even with other more complex ones.
Dirac’s equation is considered one of the most beautiful mathematical expressions in Physics. Although its form is quite simple, its development and scope have great implications.
Considerations for solving first degree equations
While each type of equation has a particular method or procedure for approaching and solving it, linear equations represent the beginning of understanding the basic rules and hierarchies during a clear. Here are some general steps to solve these types of equations:
1. If there are parentheses, these signs must be eliminated by solving the operations using the distributive property. Parentheses are removed from innermost to outermost. For example:
\(3\left( {4x – 2\left( {8 + 5x} \right) + 7} \right) = 9\)
\(3\left( {4x – 16 – 10x + 7} \right) = 9\)
\(12x – 48 – 30x + 21 = 9\)
2. In the event that it is a rational function or there are denominators, they should try to eliminate them. Both members of the equality can be multiplied by the expression of the denominator to simplify it.
3. Group the terms that contain the variable on one side of the equality and on the other side those that do not have variables. Continuing with the previous example:
\(12x – 30x = 9 – 21 + 48\)
4. Carry out the operations between similar terms in order to obtain a single term with the variable:
\( – 18x = 36\)
5. Clear the unknown through the product rule:
\(x = – \frac{{36}}{{18}}\)
6. If the result obtained is a fraction, if possible, simplify this expression:
\(x = – 2\)
Following