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In solving mathematical equations, particularly linear simultaneous equations, differential equations and integral equations, the terminology homogeneous is often used for equations with some linear operator L on the LHS and 0 on the RHS. In contrast, an equation with a non-zero RHS is called inhomogeneous or non-homogeneous, as exemplified by ...
In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign. When seeking a solution, one or more variables are designated as unknowns. A solution is an assignment of ...
Common to the class is the nature of the resulting equation, which is a linear Diophantine equation in two unknowns. Most members of the class are determinate, but some are not (the monkey and the coconuts is one of the latter). Familiar algebraic methods are unavailing for solving such equations.
Each side of the equation corresponds to one side of the balance. Different quantities can be placed on each side: if the weights on the two sides are equal, the scale balances, and in analogy, the equality that represents the balance is also balanced (if not, then the lack of balance corresponds to an inequality represented by an inequation).
[27] [28] Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. [29] Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. [30]
We can reduce the fractions to lowest terms by noting that the two occurrences of b on the left-hand side cancel, as do the two occurrences of d on the right-hand side, leaving =, and we can divide both sides of the equation by any of the elements—in this case we will use d —getting =.
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