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Linear equations can have one 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 do not include exponents.
Linear equations in two variables
A common form of a linear equation in the two variables x and y is
where m and b designate constants (parameters). 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 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 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.
- where 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.
- where 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, 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).
- where (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:
- where 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/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.
Using the order of the standard form
one can rewrite the equation in matrix form:
Further, this representation extends to systems of linear equations.
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 system of linear equations, like Gauss-Jordan which can be expressed as matrix elementary row operations.
- 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 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:
- The 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 (signed) length of the normal segment. The normal form can be derived from the general form by dividing all of the coefficients by
- This form is also called the Hesse standard form, after the German mathematician Ludwig Otto Hesse.
- Unlike the slope-intercept and intercept forms, this form can represent any line but also requires only two finite parameters, θ and p, to be specified. Note that if the line is through the origin (C=0, p=0), one drops the |C|/-C term to compute sinθ and cosθ
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.
To expand on this we can say that , and . Thus and , then the above equation becomes:
Then dividing both side by would result in the “Two-point form” shown above, but leaving it here allows the equation to still be valid when .
- This 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.
Connection with linear functions
A linear equation, written in the form y = f(x) whose graph crosses through the origin, that is, whose y-intercept is 0, has the following properties:
where a is any scalar. A function which satisfies these properties is called a linear function (or linear operator, or more generally a linear map). However, linear equations that have non-zero y-intercepts, when written in this manner, produce functions which will have neither property above and hence are not linear functions in this sense. They are known as affine functions.
Linear equations in more than two variables
A linear equation can involve more than two variables. The general linear equation in n variables is:
In this form, a1, a2, …, an are the coefficients, x1, x2, …, xn are the variables, and b is the constant. When dealing with three or fewer variables, it is common to replace x1 with just x, x2 with y, and x3 with z, as appropriate.
In vector notation, this can be expressed as:
where is a vector normal to the plane, are the coordinates of any point on the plane, and are the coordinates of the origin of the plane.
- Quadratic equation (degree = 2)
- Cubic equation (degree = 3)
- Quartic equation (degree = 4)
- Quintic equation (degree = 5)
- Barnett, Ziegler & Byleen 2008, pg. 15
- Barnett, R.A.; Ziegler, M.R.; Byleen, K.E. (2008), College Mathematics for Business, Economics, Life Sciences and the Social Sciences (11th ed.), Upper Saddle River, N.J.: Pearson, ISBN 0-13-157225-3
- Algebraic Equations at EqWorld: The World of Mathematical Equations.
-  Video tutorial on solving one step to multistep equations
- Linear Equations and Inequalities Open Elementary Algebra textbook chapter on linear equations and inequalities.
- Hazewinkel, Michiel, ed. (2001), "Linear equation", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4