Linear Equations and Functions
In mathematics, a linear equation or linear system is an equation or system of equations in which each term is linear in the others. For example, if we have the three linear equations x + 3 y + 3 z = -1, x + 3 y + 5 z = -1, and x + y + z = -1, then they are linearly dependent and therefore cannot be solved simultaneously. In contrast, x + y + z = 1 is a linear equation. If x + y + z = 1 is a linear equation and x + y + z = -1, then we have found a linear equation which is a solution to one of the linear equations. The simplest definition of a function is a rule for assigning to each element in a set a unique element of a set of the same size. For example, for each integer n, there is a function that assigns to each n a nonnegative integer. Such a function is called an integer function or a natural number function. If the integers are taken as the set of all nonnegative integers, then the set of all integer functions forms a field. In mathematical analysis, the field of study of functions is one of the basic fields of study. The field of study of functions can be extended to include fields of real-valued functions, complex-valued functions, and functions of several variables. The field of study of functions is a subfield of the field of study of functions of one variable. In calculus, a function is a rule which assigns to each element of a set a unique element of a set of the same size. For example, the function (x + y) = x + y + 2 (x + y + 3) = x + y + 2 (x + y + 3) assigns to each pair of real numbers a real number. The function (x + y) = x + y + 2 (x + y + 3) is called a composition of functions, and its graph is the composition of the graphs of the functions x + y and x + y + 3. The function (x + y) = x + y + 2 (x + y + 3) is a function of a single variable. The differentiation of a function is the process of finding the derivative of a function. If the function is given by an equation, then the derivative of that function is the solution to the associated system of differential equations, or the solution to the associated homogeneous difference equation. If the function is given by a formula, then the derivative of that function is the limit of a sequence of functions as the sequence approaches the function from the left, or the limit of a sequence of functions as the sequence approaches the function from the right, or the derivative of the function.