Definition of Linear Equations

A linear equation is a polynomial equation of the first degree. In other words, it 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.

Simplest Definition

The simplest definition of a linear equation is as a relationship between two variables that, when graphed against Cartesian axes, produces a straight line.

Simplest Form

The simplest definition's form is y=mx+b. The "m" and the "b" are the constants. Represented on the x-y plane, "m" refers to the slope of the equation, while "b" refers to the point where the line of the equation hits the y-axis.

More General Form

Linear equations are not generally defined to constrain the equation to only two variables. For example, w+x+y+z=3 is a linear equation as well. This equation x+xy+z=3 however is not, because the xy terms is second-order, not first-order.

System of Linear Equations

One can form a system of linear equations that must be solved simultaneously. For example:
3x+2y=5
3x-2y=0
The solution to this system, graphically, is the point where the two lines cross in the x-y plane.

Matrix Form of System of Equations

In linear algebra, a linear equation is defined as having the form Ax=b, b is a vector of constants (b1,b2,...), A is a matrix of coefficients, and x is a vector of variables (x1,x2,...). For example, in the prior section b=(5,0), the variable vector is (x,y) and A is
3__2
2__-2

References

• Mathematics Dictionary; James & James; 1976