Equacions

Equacions are formal statements that assert the equality of two expressions. They typically include one or more variables and constants and describe a condition that must be satisfied for certain values. A typical example is 3x + 2 = 11, and solving equacions involves finding the values of the unknowns that make the equality true.

Equacions are classified by the form of their expressions. Linear equacions have variables to the first power and no variable products, such as ax + b = c, or systems like a1x + b1y = c1 and a2x + b2y = c2. Polynomial equacions involve powers of variables up to a fixed degree. Differential equacions relate unknown functions to their derivatives; integral and functional equacions involve integrals or unspecified functions. Systems of equacions use several equations to constrain several unknowns.

Solve methods include algebraic manipulation (isolating the unknown with inverse operations), factoring, completing the square, and substitution or elimination for systems. For nonlinear equacions, numerical methods such as Newton-Raphson are used. Graphical methods show solutions as intersections of curves. Equacions model relationships in science, engineering, economics, and computer science.

Historically, solving equacions dates to ancient mathematics. The equals sign was introduced by Robert Recorde in 1557. Developments in algebra, notation, and mathematical analysis came from various cultures, enabling systematic methods for solving equacions. Today, computer algebra systems automate many steps of symbolic and numerical solving.