Linear equation

Linear equation

Video: Introduction to Linear Equations

A linear equation is an equation that may be put in the form

where are the variables or unknowns, and are coefficients, which are often real numbers, but may be parameters, or even any expression that does not contain the unknowns. In other words, a linear equation is obtained by equating to zero a linear polynomial.

The solutions of such an equation are the values that, when substituted to the unknowns, make the equality true.

The case of one unknown is of a particular importance, and it is frequent that linear equation refers implicitly to this particular case, that is to an equation that may be written in the form

If a ≠ 0 this linear equation has the unique solution

The solutions of a linear equation in two variables form a line in the Euclidean plane, and every line may be defined as the solutions of a linear equation. This is the origin of the term linear for qualifying this type of equations. More generally, the solutions of a linear equation in n variables form a hyperplane (of dimension n – 1) in the Euclidean space of dimension n.

Linear equations occur frequently in all mathematics and their applications in physics and engineering, partly because non-linear systems are often well approximated by linear equations.

One variable

Video: Solving linear equations with one variable

A linear equation in one unknown x may always be rewritten

If a ≠ 0, there is a unique solution

If a = 0, then, if b = 0, every number is a solution of the equation, and, if b ≠ 0, there are no solutions (and the equation is said to be inconsistent).

Two variables

Video: Example: Evaluating expressions with 2 variables 

A common linear equation in two variables x and y is the relation that links the argument and the value of a linear function:

where m and are real numbers. The graph of such a linear function is thus the set of the solutions of this linear equation, which is a line in the Euclidean plane of slope m and y-intercept  .

Every linear equation in x and y may be rewritten

where a and b are not both zero. The set of the solutions form a line in the Euclidean plane, which is the graph of a linear function if and only if b ≠ 0.

Using the laws of elementary algebra, linear equations in two variables may be rewritten in several standard forms that are described below, which are often referred to as "equations of a 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^[1]) form the linear equation is written as:

where A and B are not both equal to zero. The equation is ........

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