Equation

An equation is a mathematical formula that expresses the equality of two expressions. It acts as a formal statement that the left-hand side and the right-hand side possess the same value. This relationship is established by connecting the two expressions with an equals sign (=). You can think of an equation as being analogous to a weighing scale or a balance.

If you place equal weights into the two pans, the scale remains in perfect balance. If you change one side, the equality is broken unless you make the exact same change to the other side.

To maintain this balance, mathematicians follow specific rules of transformation. An equation remains equivalent if you add or subtract the same quantity from both sides. You can also multiply or divide both sides by any non-zero quantity. These operations allow you to transform a complex equation into a simpler, equivalent version. For example, you can subtract the right-hand side from both sides to create an equation where the right side is zero. This is a common technique used to simplify mathematical problems. However, one must be careful when applying functions to both sides. Applying a function can sometimes introduce extraneous solutions, which are extra answers that do not actually satisfy the original equation.

Equations are categorized into two primary types: identities and conditional equations. An identity is an equation that remains true for every possible value of the variables it contains. For instance, the difference of two squares, such as (x - y)(x + y) = x² - y², is a mathematical identity. In contrast, a conditional equation is only true for specific, particular values. Solving a conditional equation means determining which values for the unknown variables make the equality true. These values are called the solutions of the equation.

In a polynomial equation, the solutions can often be seen as the points where a graph crosses an axis.

Mathematical expressions within equations often contain variables and parameters. Variables, or unknowns, are the values we are trying to find, such as x, y, or z. Parameters are other terms that are assumed to be known or fixed within a specific context. For example, in the equation for a circle, a parameter might represent the radius.

Using these parameters, we can write a general equation that describes an entire family of shapes. Often, coefficients are denoted by letters at the beginning of the alphabet, like a or b, while unknowns use letters from the end of the alphabet.

History shows that humans have developed methods for handling equations for centuries. In 1557, a man named Robert Recorde introduced the modern equals sign.

He chose two parallel straight lines of equal length because he believed nothing could be more equal than those lines. Long before Recorde, other civilizations were mastering these concepts. An anonymous 2nd-century Chinese text titled The Nine Chapters on the Mathematical Art proposed methods for resolving linear equations.

These early mathematical traditions laid the groundwork for the complex algebra used today.

Algebraic study often focuses on polynomial equations, which can be univariate or multivariate. A univariate equation involves only one variable, while a multivariate equation involves several, such as x, y, and z. Some polynomial equations can be solved using algebraic expressions, but others cannot. The Abel–Ruffini theorem demonstrates that equations of degree five or higher cannot always be solved this way. Beyond single equations, mathematicians often work with a system of equations. A system is a collection of equations that must all be satisfied simultaneously.

Finding the common solution to a system is a fundamental task in many scientific fields.

Equations serve as a vital bridge between different branches of mathematics. In analytic geometry, equations are used to characterize geometric figures within a coordinate system. For example, an equation can define the exact path of a line or the curve of a circle. In physics, engineering, and economics, equations are used to create mathematical models of complex systems. By using linear systems, scientists can often approximate very complicated non-linear behaviors. This ability to turn physical relationships into logical formulas is what makes equations so powerful in understanding the universe.