Linear equation in the context of "Equation"

Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.

Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called systems of linear equations. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions.

Muhammad ibn Musa al-Khwarizmi, or simply al-Khwarizmi (c. 780 – c. 850) was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in Baghdad, the contemporary capital city of the Abbasid Caliphate. One of the most prominent scholars of the period, his works were widely influential on later authors, both in the Islamic world and Europe.

His popularizing treatise on algebra, compiled between 813 and 833 as Al-Jabr (The Compendious Book on Calculation by Completion and Balancing), presented the first systematic solution of linear and quadratic equations. One of his achievements in algebra was his demonstration of how to solve quadratic equations by completing the square, for which he provided geometric justifications. Because al-Khwarizmi was the first person to treat algebra as an independent discipline and introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation), he has been described as the father or founder of algebra. The English term algebra comes from the short-hand title of his aforementioned treatise (الجبر Al-Jabr, transl. "completion" or "rejoining"). His name gave rise to the English terms algorism and algorithm; the Spanish, Italian, and Portuguese terms algoritmo; and the Spanish term guarismo and Portuguese term algarismo, all meaning 'digit'.

Linear algebra is the branch of mathematics concerning linear equations such as

linear maps such as

In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables.For example,

is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. In the example above, a solution is given by the ordered triple${\displaystyle (x,y,z)=(1,-2,-2),}$since it makes all three equations valid.

In mathematics, a quadratic equation (from Latin quadratus 'square') is an equation that can be rearranged in standard form as$${\displaystyle ax^{2}+bx+c=0\,,}$$where the variable ⁠${\displaystyle x}$⁠ represents an unknown number, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear, not quadratic.) The numbers a, b, and c are the coefficients of the equation and may be distinguished by respectively calling them, the quadratic coefficient, the linear coefficient and the constant coefficient or free term.

The values of ⁠${\displaystyle x}$⁠ that satisfy the equation are called solutions of the equation, and roots or zeros of the quadratic function on its left-hand side. A quadratic equation has at most two solutions. If there is only one solution, one says that it is a double root. If all the coefficients are real numbers, there are either two real solutions, or a real double root, or two complex solutions that are complex conjugates of each other. A quadratic equation always has two roots, if complex roots are included and a double root is counted for two. A quadratic equation can be factored into an equivalent equation $${\displaystyle ax^{2}+bx+c=a(x-r)(x-s)=0}$$where r and s are the solutions for ⁠${\displaystyle x}$⁠.

In linear algebra, a linear relation, or simply relation, between elements of a vector space or a module is a linear equation that has these elements as a solution.

More precisely, if ${\displaystyle e_{1},\dots ,e_{n}}$ are elements of a (left) module M over a ring R (the case of a vector space over a field is a special case), a relation between ${\displaystyle e_{1},\dots ,e_{n}}$ is a sequence ${\displaystyle (f_{1},\dots ,f_{n})}$ of elements of R such that

The Brāhma-sphuṭa-siddhānta ("Correctly Established Doctrine of Brahma", abbreviated BSS)is a main work of Brahmagupta, written c. 628. This text of mathematical astronomy contains significant mathematical content, including the first good understanding of the role of zero, rules for manipulating both negative and positive numbers, a method for computing square roots, methods of solving linear and quadratic equations, rules for summing series, Brahmagupta's identity, and Brahmagupta theorem.

The book was written completely in verse and does not contain any kind of mathematical notation. Nevertheless, it contained the first clear description of the quadratic formula (the solution of the quadratic equation).