Definition

Linear Equations:

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 abundantly in most subareas of mathematics and especially 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.

This article considers the case of a single equation for which one searches the real solutions. All its content applies for complex solutions and, more generally for linear equations with coefficients and solutions in any field.

Graph:

In mathematics, and more specifically in graph theory, a graph is a representation of a set of objects where some pairs of objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges. Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics.

The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this is an undirected graph, because if person A shook hands with person B, then person B also shook hands with person A. In contrast, if there is an edge from person A to person B when person A knows of person B, then this graph is directed, because knowledge of someone is not necessarily a symmetric relation (that is, one person knowing another person does not necessarily imply the reverse; for example, many fans may know of a celebrity, but the celebrity is unlikely to know of all their fans). This latter type of graph is called a directed graph and the edges are called directed edges or arcs.

Vertices are also called nodes or points, and edges are also called arcs or lines. Graphs are the basic subject studied by graph theory.

Matrix:

A matrix is a rectangular array of numbers or other mathematical objects, for which operations such as addition and multiplication are defined. Each element of a matrix is often denoted by a variable with two subscripts. For instance, a_2,1 represents the element at the second row and first column of a matrix A.

 Rotation:

A transformation in which a plane figure turns around a fixed center point. In other words, one point on the plane, the center of rotation, is fixed and everything else on the plane rotates about that point by a given angle.


Translation:

A translation "slides" an object a fixed distance in a given direction.  The original object and its translation have the same shape and size, and they face in the same direction.  A translation creates a figure that is congruent with the original figure and preserves distance (length) and orientation (lettering order).  A translation is a direct isometry.

Properties preserved (invariant) under a translation:

1.  distance (lengths of segments are the same)

2.  angle measures (remain the same)

3.  parallelism (parallel lines remain parallel)

4.  colinearity (points stay on the same lines)

5.  midpoint (midpoints remain the same in each figure)

6.  orientation (lettering order remains the same