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Linear equation
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 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
\( y = mx + b,\, \)
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 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 form
\( Ax + By + C = 0, \, \)
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.
Standard form
\( Ax + By = C,\, \)
where A and B are not both equal to zero, A, B, and C are coprime integers, and A is nonnegative (if zero, B must 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. It is worth noting that, while the term occurs frequently in school-level US algebra textbooks, most lines cannot be described by such equations. For instance, the line x + y = √2 cannot be described by a linear equation with integer coefficients since √2 is irrational.
Slope–intercept form
\( y = mx + b,\, \)
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 originates at (0, b) and extends outward at a slope of m. Vertical lines, having undefined slope, cannot be represented by this form.
Point–slope form
\( y - y_1 = m( x - x_1 ),\, \)
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 - 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).
Two-point form
\( y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1),\, \)
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 point-slope form above, where the slope is explicitly given as \( \frac{y_2 - y_1}{x_2 - x_1}. \)
Intercept form
\( \frac{x}{a} + \frac{y}{b} = 1,\, \)
where 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 form
\( x = T t + U\, \)
and
\( y = V t + W.\, \)
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:
\( x = (p - h) t + h\, \)
and
\( y = (q - k)t + k.\, \)
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.
Polar form
\( r=\frac{mr\cos\theta+b}{\sin\theta}, \)
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:
\( r\sin\theta=mr\cos\theta+b.\, \)
Normal form
The normal for a given line is defined to be the shortest segment between the line and the origin. The normal form of the equation of a straight line is given by:
\( y \sin \theta + x \cos \theta - p = 0,\, \)
where θ is the angle of inclination of the normal, and p is the length of the normal. The normal form can be derived from general form by dividing all of the coefficients by
\( \frac{|C|}{-C}\sqrt{A^2 + B^2}. \)
This form is also called the Hesse standard form, after the German mathematician Ludwig Otto Hesse.
2D vector determinant form
The equation of a line can also be written as the determinant of two vectors. If P_1 and P_2 are unique points on the line, then P will also be a point on the line if the following is true:
\( \det( \overrightarrow{P_1 P} , \overrightarrow{P_1 P_2} ) = 0. \)
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 determinate equals zero then the parallelogram has no area, and that will happen when to vectors are on the same line.
To expand on this we can say that \( P_1 = (x_1 ,\, y_1), P_2 = (x_2 ,\, y_2) \) and \( P = (x ,\, y) \) . Thus \( \overrightarrow{P_1 P} = (x-x_1 ,\, y-y_1) \) and \( \overrightarrow{P_1 P_2} = (x_2-x_1 ,\, y_2-y_1) \) , then the above equation becomes:
\( \det \begin{pmatrix}x-x_1&y-y_1\\x_2-x_1&y_2-y_1\end{pmatrix} = 0. \)
Thus,
\( ( x - x_1 )( y_2 - y_1 ) - ( y - y_1 )( x_2 - x_1 )=0. \)
Ergo,
\( ( x - x_1 )( y_2 - y_1 ) = ( y - y_1 )( x_2 - x_1 ). \)
Then dividing both side by \( ( x_2 - x_1 ) \) would result in the “Two-point form” shown above, but leaving it here allows the equation to still be valid when \( x_1 = x_2. \)
Special cases
\( y = b\, \)
Horizontal Line y = b
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.
\( x = a\, \)
Vertical Line x = a
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.
\( y = y \ and x = x\, \)
In this case all variables and constants have canceled out, leaving a trivially true statement. The original equation, therefore, would be called an identity 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.
\( e = f\, \)
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
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:
\( f ( x_1 + x_2 ) = f ( x_1) + f ( x_2 )\ \)
and
\( f ( a x ) = a f ( x ),\, \)
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 will have neither property above and hence are not linear functions in this sense.
Linear equations in more than two variables
Main article: System of linear equations
A linear equation can involve more than two variables. The general linear equation in n variables is:
\( a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b. \)
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.
Such an equation will represent an (n–1)-dimensional hyperplane in n-dimensional Euclidean space (for example, a plane in 3-space).
In vector notation, this can be expressed as:
\( \overrightarrow{n} \cdot \overrightarrow{x} = \overrightarrow{n} \cdot \overrightarrow{x_0} \)
where \( \overrightarrow{n} \) is a vector normal to the plane, \( \overrightarrow{x} \) are the coordinates of any point on the plane, and \( \overrightarrow{x_0} \) are the coordinates of the origin of the plane.
See also
Line (geometry)
Quadratic equation
Cubic equation
Quartic equation
Quintic equation
Linear inequality
Linear belief function
References
External links
Algebraic Equations at EqWorld: The World of Mathematical Equations.
[1] Video tutorial on solving one step to multistep equations
Linear Equations and Inequalities Open Elementary Algebra textbook chapter on linear equations and inequalities.
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