Equation

An equation is a way to show when two things are the same value in mathematics. It uses a mathematical formula with an equals sign =. This sign means the value on one side is the same as the value on the other side. Equations help us see how numbers and values are related. We use them in many places, from simple math to big science ideas.

Solving an equation means finding which numbers make it true. These numbers are called solutions. There are two main kinds of equations. Identities are true for every number you try. Other equations are true only for some special numbers.

The equals sign "=" was created in 1557 by Robert Recorde. He picked it because he thought two lines that are the same length and parallel could not be more equal. Equations are useful for solving problems and finding 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 the same if they have the same answers. We can change an equation into another one with the same answers by doing these things:

• 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 math rules to change one side, like spreading out 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 answers that weren’t there before. For example, the equation x = 1 becomes x² = 1 after we square both sides. This new equation has two answers: x = 1 and x = -1. We need to 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 we already know, called constants, and others we need to find, called unknowns. For example, in the equation x² + y² = R², R is a number we know. If R is 2, this equation describes a circle.

An identity is an equation that is always true, no matter what numbers you use. For example, x² − y² = (x + y)(x − y) is true for any x and y. Identities help solve harder equations.

Algebra

Algebra studies two main types of equations: polynomial equations and linear equations. Polynomial equations look like P(x) = 0, where P is a polynomial. Linear equations look like ax + b = 0, where a and b are numbers.

To solve these, mathematicians use ideas 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.

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 studies special kinds of equations and their answers. Diophantine equations search for whole number answers, named after the ancient mathematician Diophantus. These equations can be easy or very hard, and solving them means finding which whole numbers work for all parts of the equation.

Algebraic numbers are answers to polynomial equations with rational numbers, while numbers that aren’t answers to such equations, like π, are called transcendental. Algebraic geometry looks at the shapes and patterns that come from solving groups of polynomial equations, studying special points and how these shapes connect.

Differential equations

Main article: Differential equation

A differential equation is a special kind of mathematical statement. It shows how a number changes, called its rate of change or derivative. These equations help us understand how things change over time or space. They are useful in many areas like physics, chemistry, and biology to describe things such as how heat spreads or how groups of living things grow.

In math, people study differential equations to find their solutions—the functions that work with the equations. Some easier equations can be solved with formulas. Others need special methods or computers to get close answers. There are many types of differential equations, each good for different kinds of real-world problems.

Types of equations

Equations can be grouped by the kind of math they use. Some important types are:

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

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