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In the simple case of a function of one variable, say, h(x), we can solve an equation of the form h(x) = c for some constant c by considering what is known as the inverse function of h. Given a function h : A → B , the inverse function, denoted h −1 and defined as h −1 : B → A , is a function such that
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The above equations are efficient to use if the mean of the x and y variables (¯ ¯) are known. If the means are not known at the time of calculation, it may be more efficient to use the expanded version of the α ^ and β ^ {\displaystyle {\widehat {\alpha }}{\text{ and }}{\widehat {\beta }}} equations.
The phrase "linear equation" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line. If b ≠ 0 , the line is the graph of the function of x that has been defined in the preceding section.
One obtains the value f(r) by substitution of the value r for the symbol x in P. One reason to distinguish between polynomials and polynomial functions is that, over some rings, different polynomials may give rise to the same polynomial function (see Fermat's little theorem for an example where R is the integers modulo p).
The immediate execution mode of operation (also known as single-step, algebraic entry system (AES) [7] or chain calculation mode) is commonly employed on most general-purpose calculators. In most simple four-function calculators, such as the Windows calculator in Standard mode and those included with most early operating systems, each binary ...
In the first equation, solve for one of the variables in terms of the others. Substitute this expression into the remaining equations. This yields a system of equations with one fewer equation and unknown. Repeat steps 1 and 2 until the system is reduced to a single linear equation.
Consider a system of n linear equations for n unknowns, represented in matrix multiplication form as follows: = where the n × n matrix A has a nonzero determinant, and the vector = (, …,) is the column vector of the variables. Then the theorem states that in this case the system has a unique solution, whose individual values for the unknowns ...
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