Linear Equations in Two Variables
Linear equations in two variables
A common form of a linear equation in the two variables x and y isSince 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, x^{2}, y^{1/3}, and sin(x) are nonlinear.
Forms for 2D linear equations
Linear 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) form
 In the general (or standard^{}) 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 xintercept, that is, the xcoordinate of the point where the graph crosses the xaxis (where, y is zero), is C/A. If B is nonzero, then the yintercept, that is the ycoordinate of the point where the graph crosses the yaxis (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 form
 where m is the slope of the line and b is the yintercept, which is the ycoordinate 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 form
 where m is the slope of the line and (x_{1},y_{1}) is any point on the line.
 The pointslope form expresses the fact that the difference in the y coordinate between two points on a line (that is, y − y_{1}) is proportional to the difference in the x coordinate (that is, x − x_{1}). The proportionality constant is m (the slope of the line).
Twopoint form

 where (x_{1}, y_{1}) and (x_{2}, y_{2}) are two points on the line with x_{2} ≠ x_{1}. This is equivalent to the pointslope form above, where the slope is explicitly given as (y_{2} − y_{1})/(x_{2} − x_{1}).
Intercept form

 where a and b must be nonzero. The graph of the equation has xintercept a and yintercept b. The intercept form is in standard form with A/C = 1/a and B/C = 1/b. Lines that pass through the origin or which are horizontal or vertical violate the nonzero condition on a or b and cannot be represented in this form.
Matrix form
Using the order of the standard formParametric form

 and
 Two simultaneous equations in terms of a variable parameter t, with slope m = V / T, xintercept (VU−WT) / V and yintercept (WT−VU) / T.
 This can also be related to the twopoint form, where T = p−h, U = h, V = q−k, 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.
2D vector determinant form
The equation of a line can also be written as the determinant of two vectors. If and are unique points on the line, then will also be a point on the line if the following is true: One way to understand this formula is to use the fact that the determinant of two vectors on the plane will give the area of the parallelogram they form. Therefore, if the determinant equals zero then the parallelogram has no area, and that will happen when two vectors are on the same line.
Special cases


 This is a special case of the standard form where A = 0 and B = 1, or of the slopeintercept form where the slope m = 0. The graph is a horizontal line with yintercept equal to b. There is no xintercept, unless b = 0, in which case the graph of the line is the xaxis, and so every real number is an xintercept.


 This is a special case of the standard form where A = 1 and B = 0. The graph is a vertical line with xintercept equal to a. The slope is undefined. There is no yintercept, unless a = 0, in which case the graph of the line is the yaxis, and so every real number is a yintercept.
Connection with linear functions
A linear equation, written in the form y = f(x) whose graph crosses the origin (x,y) = (0,0), that is, whose yintercept is 0, has the following properties:For Detail Please Join Ajit Mishra's Online Classroom By CLICK HERE
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