Equation

An equation is a statement in mathematics that two expressions are equal, conventionally written with the equals sign $=$. Equations may express numerical facts, identities, or conditions involving unknown quantities. In algebra, solving an equation means finding all admissible values of its unknowns that make the equality true. For example, $2x+3=11$ has the solution $x=4$. An equation differs from an expression such as $2x+3$, which represents a quantity without asserting an equality. (openstax.org)

Meaning, notation, and solutions

The expressions before and after the equals sign are called the left-hand side and right-hand side. Letters can represent unknowns, varying quantities, or parameters treated as fixed within a particular problem. A solution set contains every admissible assignment that satisfies the equation; with several unknowns, solutions are usually ordered pairs, triples, or longer tuples. (openstax.org)

The domain of admissible values matters. For example, $x^2=2$ has two solutions among the real numbers but none among the integers. The equation $x^2+1=0$ has no real solution, whereas among the complex numbers its solutions are $i$ and $-i$, where $i^2=-1$. Expressions must also be defined: an equation containing $1/x$ excludes $x=0$. (openstax.org)

An identity holds for every value in its stated domain, as in $(x+1)^2=x^2+2x+1$. A conditional equation holds only for certain values, while an inconsistent equation has no solution in the specified domain. Over the real numbers, $x+1=x$ is inconsistent. This distinction separates equalities used as general algebraic rules from equalities imposing restrictions on unknowns. (openstax.org)

Principal types

A linear equation in one unknown can be written

and has the solution $x=-b/a$. In several unknowns, linear equations contain a sum of constant multiples of the unknowns, possibly with a constant term, but no products between unknowns or higher powers of them. (openstax.org)

A quadratic equation has the form

Its solutions are given by

For real coefficients, the discriminant $b^2-4ac$ determines whether there are two distinct real solutions, one repeated real solution, or two nonreal complex solutions. More generally, a polynomial equation sets a polynomial expression equal to zero. (openstax.org)

Other classifications depend on the operations appearing in the equation. Rational equations contain quotients of polynomials; radical equations contain unknowns under root signs. Exponential, logarithmic, and trigonometric equations involve the corresponding functions. Their domains and solution methods differ, so superficially similar equations need not behave alike. (openstax.org)

A differential equation relates an unknown function to one or more of its derivatives. Its solutions are functions rather than merely numbers. For instance, $y'=y$ has the family $y=Ce^x$; an additional condition such as $y(0)=2$ selects $y=2e^x$. Initial conditions and questions of existence and uniqueness are central to this subject. (ocw.mit.edu)

Equivalent transformations

Algebraic solution methods transform an equation into simpler equations with the same solution set. Adding or subtracting the same defined quantity on both sides preserves equality. Multiplying or dividing both sides by a known nonzero quantity is also reversible. Thus $2x+3=11$ becomes $2x=8$, then $x=4$. (openstax.org)

Not every operation preserves equivalence. Squaring both sides can introduce additional candidates: $x=-2$ implies $x^2=4$, but the latter also permits $x=2$. Dividing by an expression involving an unknown can lose solutions if that expression is zero. Clearing denominators requires retaining the original exclusions. Candidate answers must therefore satisfy the original equation, not merely a transformed version. (openstax.org)

Factoring provides another method. Over the real or complex numbers, a product is zero only if at least one factor is zero. Hence

gives $x=2$ or $x=3$. This principle underlies many elementary polynomial-solving procedures. (openstax.org)

Systems and geometric interpretation

A system consists of equations that must hold simultaneously. In a system of linear equations, a solution satisfies every equation. For example,

has the unique solution $(x,y)=(3,2)$. Substitution and elimination are standard methods; larger systems can be organized using a matrix and studied within linear algebra. (openstax.org)

In geometry, equations describe sets of points. Two linear equations in two real variables represent lines, and their common solutions are intersection points. Distinct parallel lines give no solution; intersecting lines give one; coincident lines give infinitely many. This connects algebraic consistency with geometric relationships. (openstax.org)

Numerical methods and modeling

When exact solutions are difficult to obtain, an algorithm can approximate them. Newton’s method uses the iteration

to seek a root of $f(x)=0$. It draws on calculus, requires a nonzero derivative at each step, and may fail to converge depending on the function and starting value. (openstax.org)

Equations also encode models of changing systems in physics and engineering. Differential equations describe oscillations, damping, electrical circuits, and responses to external inputs. Their solutions depend on both the equation and the accompanying conditions. (ocw.mit.edu)

Historical notation

Written equations preceded the modern equals sign. The Welsh mathematician Robert Recorde introduced $=$ in The Whetstone of Witte, published in 1557, choosing parallel line segments to represent equality. The symbol was not immediately universal; competing notations remained in use into the eighteenth century. (mathshistory.st-andrews.ac.uk)