Equation

In mathematics, an equation is a mathematical formula that shows the equality of two expressions, connected by the equals sign =. Equations help us understand how different numbers and values relate to each other. They are used in many areas, from simple arithmetic to complex scientific calculations.

Solving an equation means finding the values of the variables that make the equality true. These values are called solutions. There are two main types of equations: identities, which are true for all values, and conditional equations, which are true only for specific values.

The symbol "=" was invented in 1557 by Robert Recorde. He chose it because he believed nothing could be more equal than two parallel straight lines of the same length. Equations are important tools for solving problems and exploring patterns in numbers and shapes.

Description

An equation is a math sentence that shows that two things are equal. It is written with two expressions on either side of an equals sign (=). The side on the left is called the "left-hand side," and the side on the right is called the "right-hand side."

A common type of equation is called a polynomial equation or algebraic equation. In these equations, each side is made up of one or more parts called terms. For example, in the equation

A x² + B x + C − y = 0,

the left-hand side has four terms: A x², B x, C, and −y. The right-hand side has just one term, which is 0.

Equations are like a balance scale. If you have the same weight on both sides, the scale stays balanced. If you take something away from one side, you must take the same thing away from the other side to keep it balanced. The same idea works with equations—if you do the same operation to both sides, the equation stays true.

Properties

Two equations are considered the same if they have the same solutions. We can change an equation into another one with the same solutions by doing the following:

• Adding or subtracting the same number from both sides of the equation
• Multiplying or dividing both sides by a number that is not zero
• Using mathematical rules to change one side, like expanding a product or factoring a sum
• For a group of equations: adding one equation to another after multiplying it by a number

Sometimes, when we use a special math rule on both sides, we might get extra solutions that weren't there before. For example, the equation x = 1 becomes x² = 1 after we square both sides. This new equation has two solutions: x = 1 and x = -1. We must be careful when changing equations this way.

These ideas help us solve equations using simple methods and more advanced ones like Gaussian elimination.

Examples

An equation is like a weighing scale or seesaw. Each side of the equation is like one side of the balance. If the weights on both sides are equal, the scale balances, just like when an equation is true.

Equations often have numbers that we already know, called constants or parameters, and others we need to find, called unknowns. For example, in the equation x² + y² = R², R is a parameter. If R is 2, this equation describes a circle.

An identity is an equation that is always true, no matter what values the variables have. For example, x² − y² = (x + y)(x − y) is true for any x and y. Identities help solve more complicated equations.

Algebra

Algebra studies two main families of equations: polynomial equations and linear equations. Polynomial equations look like P(x) = 0, where P is a polynomial. Linear equations have the form ax + b = 0, where a and b are numbers called parameters. To solve these equations, mathematicians use methods from linear algebra or mathematical analysis.

Algebra also looks at Diophantine equations, where the numbers used and the answers must all be integers. These are studied using ideas from number theory and can be very tricky to solve. Sometimes, we just want to know if a solution exists and, if so, how many solutions there are.

Geometry

Main article: Analytic geometry

In geometry, we can use coordinates to describe shapes and positions. By placing a grid around space, we can write equations that match points on shapes like lines and planes. For example, a plane in space can be shown with an equation using the coordinates x, y, and z.

We can also describe curves and other shapes using special equations. The idea of linking geometry with algebra, started by René Descartes, helps us solve many geometric problems using math. This link lets us turn shapes into equations, making it easier to study their properties. For instance, a circle can be described by an equation showing how its points relate to its center.

Main article: Parametric equation

Sometimes, we describe curves using parameters. A parameter is a special value that changes to show different points on a curve. For example, equations using a parameter t can describe a unit circle by showing how x and y change together as t changes. This method works for more complex shapes too, letting us build up descriptions for surfaces and higher-dimensional objects.

Number theory

Main article: Diophantine equation

Main articles: Algebraic number and Transcendental number

Main article: Algebraic geometry

Number theory explores special kinds of equations and their solutions. Diophantine equations look for whole number solutions to equations, named after the ancient mathematician Diophantus. These equations can be simple or very complex, and solving them means finding which whole numbers fit all parts of the equation.

Algebraic numbers are solutions to polynomial equations with rational numbers as coefficients, while numbers that aren’t solutions to such equations, like π, are called transcendental. Algebraic geometry studies the shapes and patterns that come from solving sets of polynomial equations, looking at special points and how these shapes relate to each other.

Differential equations

Main article: Differential equation

A differential equation is a special type of mathematical statement that connects a function with its rates of change, called derivatives. These equations help us understand how things change over time or space. They are important in many fields like physics, chemistry, and biology because they can describe processes such as how heat spreads or how populations grow.

In mathematics, scholars study differential equations to learn about their solutions—the functions that satisfy the equations. While some simpler equations can be solved exactly with formulas, others need approximations using computers. There are different kinds of differential equations, each useful for modeling different types of problems in the real world.

Types of equations

Equations can be grouped based on the kinds of math operations and values they use. Some important types include:

• An algebraic equation or polynomial equation, where both sides are polynomials. These are grouped further by their degree, such as linear equation (degree one), quadratic equation (degree two), and so on.
• A Diophantine equation is one where the solutions must be integers.
• A transcendental equation uses a transcendental function in its unknowns.
• A parametric equation expresses solutions as functions of other values called parameters.
• A functional equation involves functions instead of simple numbers.

Equations can also include derivatives and integrals, leading to types like differential equations, integral equations, and others.