A linear equation
is an algebraic equation
in which each term
is either a constant
or the product of a constant and (the first power of) a single variable
Linear equations can have one, two, three or more variables.
Linear equations occur with great regularity in applied mathematics. While they arise quite naturally when modeling many phenomena, they are particularly useful since many non-linear equations may be reduced to linear equations by assuming that quantities of interest vary to only a small extent from some "background" state.
Linear equations in two variables
A common form of a linear equation in the two variables and is
where m and b designate constants (the variable y is multiplied by the constant 1, which as usual is not explicitly written). 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 determines the slope or gradient of that line; and the constant term determines the point at which the line crosses the y-axis.
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.
Examples of linear equations in two variables:
Forms for 2D linear equations
Complicated linear equations, such as the ones above, 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
are variables; other letters represent constants
- 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 (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 slope of the line is −A/B.
- where 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 A or B is zero.
- where m is 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 , which immediately gives .
- where m ≠ 0, is the slope of the line and c is the x-intercept, which is the x-coordinate of the point where the line crosses the x axis. This can be seen by letting , which immediately gives .
- where 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, ) is proportional to the difference in the x coordinate (that is, ). The proportionality constant is m (the slope of the line).
- where c and b must be nonzero. The graph of the equation has x-intercept c and y-intercept b. The intercept form can be converted to the standard form by setting A = 1/c, B = 1/b and C = 1.
- where p ≠ h. The graph passes through the points (h,k) and (p,q), and has slope m = (q−k) / (p−h).
- Two simultaneous equations in 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:
- 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.
- where 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 by