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Linear equations in two variables

A common form of a linear equation in the two variables xand y is

where 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 line in the plane. In this particular equation, the constant m determines the slope or gradient of that line, and the constant term b determines the point at which the line crosses the y-axis, otherwise known as the y-intercept.

Since terms of 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.

Forms for 2D linear equationsLinear equations can be rewritten using the laws of elementary algebra into several different forms. These equations are often referred to as the "equations of the straight line." In what follows, x, y, t, and θ are variables; other letters represent constants (fixed numbers). General (or standard) formIn the general (or standard[1]) form the linear equation is written as:where A and B are not both equal to zero. The equation is usually written so that A ≥ 0, by convention. The graph of the equation is a straight line, 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-coordinate of the point where the graph crosses the x-axis (where, 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 (where x is zero), is C/B, and the slope of the line is −A/B. The general form is sometimes written as: where a and b are not both equal to zero. The two versions can be converted from one to the other by moving the constant term to the other side of the equal sign. Slope--intercept formwhere m is the slope of the line and b is the y-intercept, which is the y-coordinate of the location where line crosses the y axis. This can be seen by letting x = 0, which immediately gives y = b. It may be helpful to think about this in terms of y = b + mx; where the line passes through the point (0, b) and extends to the left and right at a slope of m. 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 form expresses the fact that the difference in the y coordinate between two points on a line (that is, yy1) is proportional to the difference in the x coordinate (that is, xx1). 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).

Multiplying both sides of this equation by (x2 − x1) yields a form of the line generally referred to as the symmetric form:

Intercept formwhere a and b must be nonzero. The graph of the equation has x-intercept a and y-intercept b. The intercept form is in standard form with A = 1/a and B = 1/b. Matrix form

Using the order of the standard form

one can rewrite the equation in matrix form:

Further, this representation extends to systems of linear equations.

becomes

Since this extends easily to higher dimensions, it is a common representation in linear algebra, and in computer programming. There are named methods for solving simultaneous linear equations like Gauss-Jordan which can be expressed as matrix elementary row operations.

Parametric formand Two simultaneous equations in terms of a variable parameter t, with slope m = V / T, x-intercept (VUWT) / V and y-intercept (WTVU) / T.This can also be related to the two-point form, where T = ph, U = h, V = qk, and W = k: and In this case t varies from 0 at point (h,k) to 1 at point (p,q), with values of t between 0 and 1 providing interpolation and 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. The equation can be rewritten to eliminate discontinuities: Normal formThe normal segment for a given line is defined to be the line segment drawn from the origin perpendicular to the line. This segment joins the origin with the closest point on the line to the origin. The normal form of the equation of a straight line is given by: where θ is the angle of inclination of the normal segment, and p is the length of the normal segment. The normal form can be derived from the general form by dividing all of the coefficients byThis form is also called the Hesse standard form,[citation needed] after the German mathematician Ludwig Otto Hesse.shawal is the best
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6y ago

Because we need them without network we cant be here on Google or in Firefox without that you can't talk to people or text them and watch Netflix or YouTube on this time we all depend on networks without that we can't even watch TV or listen to songs.

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play guildwars 2 may inprove or better smash your network modem then check is it still woking. if it still work then it is improve

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14y ago

You use networks for communication to share files.

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