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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.
In advanced mathematics texts, the term linear function often denotes specifically homogeneous linear functions, while the term affine function is used for the general case, which includes . The natural domain of a linear function f ( x ) {\displaystyle f(x)} , the set of allowed input values for x , is the entire set of real numbers , x ∈ R ...
The idea is to start with an initial guess, then to approximate the function by its tangent line, and finally to compute the x-intercept of this tangent line. This x-intercept will typically be a better approximation to the original function's root than the first guess, and the method can be iterated. x n+1 is a better approximation than x n ...
In mathematics, the term linear function refers to two distinct but related notions: [1] In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. [2] For distinguishing such a linear function from the other concept, the term affine function is often used ...
The equation of a line can be given in vector form: = + Here a is the position of a point on the line, and n is a unit vector in the direction of the line. Then as scalar t varies, x gives the locus of the line. The distance of an arbitrary point p to this line is given by
The normal form can be derived from the standard form + = by dividing all of the coefficients by +. and also multiplying through by if < Unlike the slope-intercept and intercept forms, this form can represent any line but also requires only two finite parameters, φ {\displaystyle \varphi } and p , to be specified.
Functions of the form = have at most one -intercept, but may contain multiple -intercepts. The x {\displaystyle x} -intercepts of functions, if any exist, are often more difficult to locate than the y {\displaystyle y} -intercept, as finding the y {\displaystyle y} -intercept involves simply evaluating the function at x = 0 {\displaystyle x=0} .
Starting with initial values x 0 and x 1, we construct a line through the points (x 0, f(x 0)) and (x 1, f(x 1)), as shown in the picture above.In slope–intercept form, the equation of this line is