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The simplest is the slope-intercept form: = +, from which one can immediately see the slope a and the initial value () =, which is the y-intercept of the graph = (). Given a slope a and one known value () =, we write the point-slope form:
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.
Solving quintic equations in terms of radicals (nth roots) was a major problem in algebra from the 16th century, when cubic and quartic equations were solved, until the first half of the 19th century, when the impossibility of such a general solution was proved with the Abel–Ruffini theorem.
In two dimensions, the equation for non-vertical lines is often given in the slope-intercept form: = + where: m is the slope or gradient of the line. b is the y-intercept of the line. x is the independent variable of the function y = f(x).
Slope illustrated for y = (3/2)x − 1.Click on to enlarge Slope of a line in coordinates system, from f(x) = −12x + 2 to f(x) = 12x + 2. The slope of a line in the plane containing the x and y axes is generally represented by the letter m, [5] and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line.
Suppose that two lines have the equations y = ax + c and y = bx + d where a and b are the slopes (gradients) of the lines and where c and d are the y-intercepts of the lines. At the point where the two lines intersect (if they do), both y coordinates will be the same, hence the following equality: + = +.
Power functions – relationships of the form = – appear as straight lines in a log–log graph, with the exponent corresponding to the slope, and the coefficient corresponding to the intercept. Thus these graphs are very useful for recognizing these relationships and estimating parameters .
The point-slope form of an equation forms an equation of a line, given a point (,) and slope . The general form of this equation is: y − K = M ( x − H ) {\displaystyle y-K=M(x-H)} . Using the point ( a , f ( a ) ) {\displaystyle (a,f(a))} , L a ( x ) {\displaystyle L_{a}(x)} becomes y = f ( a ) + M ( x − a ) {\displaystyle y=f(a)+M(x-a)} .