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This is a picture of a contaminant plume being emitted from a smokestack in 2-dimensional space. The smokestack and its plume are reflected across the x-axis, to account for the mass of the contaminant that bounces and (perfectly) reflects off of the boundary (the ground). Any mass that is lost from the original plume is replaced by the image.
Conversely, every line is the set of all solutions of a linear equation. The phrase "linear equation" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line. If b ≠ 0, the line is the graph of the function of x that has been defined in the preceding ...
Lines in a Cartesian plane or, more generally, in affine coordinates, are characterized by linear equations. More precisely, every line L {\displaystyle L} (including vertical lines) is the set of all points whose coordinates ( x , y ) satisfy a linear equation; that is, L = { ( x , y ) ∣ a x + b y = c } , {\displaystyle L=\{(x,y)\mid ax+by=c ...
A flat can be described by a system of linear equations.For example, a line in two-dimensional space can be described by a single linear equation involving x and y: + = In three-dimensional space, a single linear equation involving x, y, and z defines a plane, while a pair of linear equations can be used to describe a line.
This familiar equation for a plane is called the general form of the equation of the plane or just the plane equation. [6] Thus for example a regression equation of the form y = d + ax + cz (with b = −1) establishes a best-fit plane in three-dimensional space when there are two explanatory variables.
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set .
The solutions of an inhomogeneous linear differential equation form an affine space over the solutions of the corresponding homogeneous linear equation. Generalizing all of the above, if T : V → W is a linear map and y lies in its image , the set of solutions x ∈ V to the equation T x = y is a coset of the kernel of T , and is therefore an ...
Expansion of this determinantal equation produces a homogeneous linear equation, which must be the equation of line l. Therefore, up to a common non-zero constant factor we have l = [a, b, c] where: a = y 1 z 2 - y 2 z 1, b = x 2 z 1 - x 1 z 2, and c = x 1 y 2 - x 2 y 1. In terms of the scalar triple product notation for vectors, the equation ...
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