General form of standard form and the equation of the line in standard form?

Answer 1

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Linear equations in two variables

A common form of a linear equation in the two variables xand y iswhere m and b designate constants. The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a Straight_linein the plane. In this particular equation, the constant mdetermines the Slopeor gradient of that line, and the Constant_termb determines the point at which the line crosses the y-axis, otherwise known as the y-intercept.

Since terms of a linear equations cannot contain products of distinct or equal variables, nor any power (other than 1) or other function of a variable, equations involving terms such as xy, x2, y1/3, and sin(x) are Nonlinear_system.

Forms for 2D linear equations

Linear equations can be rewritten using the laws of Elementary_algebrainto several different forms. These equations are often referred to as the "equations of the straight line". In what follows x, y and t are variables; other letters represent Constant_term(fixed numbers).

General formwhere A and B are not both equal to zero. The equation is usually written so that A ≥ 0, by convention. The Cartesian_coordinate_systemof the equation is a Line_(geometry), and every straight line can be represented by an equation in the above form. If A is nonzero, then the x-intercept, that is the x-Coordinateof the point where the graph crosses the x-axis (y is zero), is −C/A. If B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis (x is zero), is −C/B, and the Slopeof the line is −A/B. Standard formwhere A, B, and C are integers whose greatest common factor is 1, A and B are not both equal to zero, and A is non-negative (and if A = 0 then B has to be positive). The standard form can be converted to the general form, but not always to all the other forms if Aor B is zero. It is worth noting that, while the term occurs frequently in school-level US textbooks, it makes little mathematical sense since most lines cannot be described by such equations. For instance, the line x + y = √2cannot be described by a linear equation with integer coefficients since √2 is irrational. Slope-intercept formwhere mis the slope of the line and b is the y-intercept, which is the y-coordinate of the point where the line crosses the y axis. This can be seen by letting x = 0, which immediately gives y = b. Vertical lines, having undefined slope, cannot be represented by this form. Point-slope formwhere m is the slope of the line and (x1,y1) is any point on the line. The point-slope and slope-intercept forms are easily interchangeable.The point-slope form expresses the fact that the difference in the y coordinate between two points on a line (that is, y − y1) is proportional to the difference in the x coordinate (that is, x − x1). The proportionality constant is m (the slope of the line). Two-point formwhere (x1,y1) and (x2,y2) are two points on the line with x2 ≠ x1. This is equivalent to the point-slope form above, where the slope is explicitly given as (y2−y1) / (x2−x1).Intercept formwhere a and b must be nonzero. The graph of the equation has x-intercept a and y-intercept b. The intercept form can be converted to the standard form by setting A = 1/a, B = 1/b and C = 1. Parametric formandTwo Simultaneous_equationsin terms of a variable parameter t, with slope m = V / T, x-intercept (VU−WT) / V and y-intercept (WT−VU) / T.This can also be related to the two-point form, where T = p−h, U = h, V = q−k, and W = k:andIn this case tvaries from 0 at point (h,k) to 1 at point (p,q), with values of t between 0 and 1 providing Interpolationand other values of t providing Extrapolation. Polar Formwhere m is the slope of the line and b is the Y-intercept. When θ = 0 the graph will be undefined. Thus, the equation can be rewritten to eliminate discontinuities: Normal formwhere φ is the angle of inclination of the normal and p is the length of the normal. The normal is defined to be the shortest segment between the line in question and the origin. Normal form can be derived from general form by dividing all of the coefficients byThis form is also called the Hesse standard form, after the German mathematician Otto_Hesse. Special casesThis is a special case of the standard form where A = 0 and B = 1, or of the slope-intercept form where the slope M = 0. The graph is a horizontal line with y-intercept equal to b. There is no x-intercept, unless b = 0, in which case the graph of the line is the x-axis, and so every real number is an x-intercept.This is a special case of the standard form where A = 1 and B = 0. The graph is a vertical line with x-intercept equal to a. The slope is undefined. There is no y-intercept, unless a = 0, in which case the graph of the line is the y-axis, and so every real number is a y-intercept.andIn this case all variables and constants have canceled out, leaving a trivially true statement. The original equation, therefore, would be called an Identity_(mathematics)and one would not normally consider its graph (it would be the entire xy-plane). An example is 2x + 4y = 2(x + 2y). The two expressions on either side of the equal sign are always equal, no matter what values are used for x and y.In situations where algebraic manipulation leads to a statement such as 1 = 0, then the original equation is called inconsistent, meaning it is untrue for any values of x and y (i.e. its graph would be the Empty_set) An example would be 3x + 2 = 3x − 5. Connection with linear functions and operatorsIn all of the named forms above (assuming the graph is not a vertical line), the variable y is a Function_(mathematics) of x, and the graph of this function is the graph of the equation.

In the particular case that the line crosses through the origin, if the linear equation is written in the form y = f(x) then f has the properties:

and

where a is any Scalar_(mathematics). A function which satisfies these properties is called a linear function, or more generally a Linear_map. This property makes linear equations particularly easy to solve and reason about. WIKIPEDIA ROCKS MAN!!!!!!!!!!!!