# Linear Equations

An equation is called linear if it can be expressed as a polynomial of degree 1 equal to 0.

## Definition of a Linear Equation

If p(x_{1}, ..., x_{n}) is a degree 1 polynomial with n unknowns x_{1}, ...,x_{n}

**a _{1}x_{1} + ... + a_{n}x_{n} -b**

where n is an integer belonging to the set of integers Z and is greater than or equal to 1

**n ∈ Z **≥** 1**

and both the coefficients a_{1},...,a_{n} and the constant term b are real numbers,

then

**a _{1}x_{1} + ... + a_{n}x_{n} -b = 0 **

is a linear equation with n unknowns.

**Note**: The simplest case is a linear equation with one unknown **a _{1}x_{1} -b = 0 ** or more simply

**a**.

_{1}x_{1}= b## Solving the linear equation

Given a linear equation with n unknowns

its solution is an ordered n-tuple of real numbers ( s_{1},..., s_{n} )

which, when substituted in place of the unknowns ( x_{1}, ..., x_{n} ), make the equation true.

## Examples of Linear Equations

The following are examples of linear equations.

**How do we recognize non-linear equations?**

An algebraic polynomial equation might not be linear for several reasons.

- If the polynomial is of a degree higher than the first.
- If the equation is not polynomial (for example, if it's transcendental).

__Examples of Non-Linear Equations__:

## Geometric Representation

A linear equation is represented in an N-dimensional space, equal to the number of unknowns.

### Example of a Linear Equation with Two Unknowns

The following linear equation has two unknown variables (x_{1},x_{2})

and can be graphically represented by a line on a Cartesian plane.

The coordinates (x_{1},x_{2}) of points on the line identify the ordered pairs of real numbers that are the possible solutions of the linear equation.

**The solutions of the linear equation are infinite**

The set of possible solutions of the linear equation is infinite, just like the points on the line.