# Consistent And Inconsistent Linear System

Some linear systems have no solution; they are called inconsistent systems. If a linear system has at least one solution than it is called consistent system.

To illustrate this with example, consider two linear equations that represents two line on xy-plane given below.

$a_{1}x{1} + a_{2}x_{2} = b_{1}$
$a_{3}x{1} + a_{4}x_{2} = b_{2}$

where either one variable can be zero, not both.

Let us call the lines as $L_{1}$ and $L_{2}$. A solution to the equation is a point $(x, y)$ that satisfy both equations given above.

### Possibilities of Solution

There are three possibilities of solution for lines – $L_{1}$ and $L_{2}$.

• No Solutions
• Exactly One Solution
• Infinite Solutions

Let us examine each one of these possibilities closely.

#### No Solutions

If the lines – $L_{1}$ and $L_{2}$ are parallel, that is, not touching each other, there is no possibility of a common point $(x,y)$.

Therefore, the above system has no solution.

#### Exactly One Solution

Suppose the lines – $L_{1}$ and $L_{2}$ intersect each other exactly at a point $(x, y)$. Then we have exactly one solution to the linear system.

The point $(x, y)$ in the above diagram is the only solution that satisfy equation of both the lines.

#### Infinite Solutions

When the two lines – $L_{1}$ and $L_{2}$ are the same or overlap each other then we can say that there could be more than one points $(x, y)$ belong to both line.

Therefore, the equation has infinite solutions because the line may increase forever in the xy-plane. We can formally say that ” Every linear system has no solution, one solution, or infinitely many solutions“.

### Augmented Matrix

We know linear equations and system of linear equations from our earlier discussions. Any system of linear equation with $m$ equations and $n$ variable $x_{1}, x_{1}, x_{1}$, …, $x_{1}$ is written as

$a_{11}x_{1} + a_{12}x_{2}+ a_{13}x_{3} = b_{1}$$a_{11}x_{1} + a_{12}x_{2}+ a_{13}x_{3} = b_{1}$
$a_{21}x_{1} + a_{22}x_{2}+ a_{23}x_{3} = b_{2}$$a_{21}x_{1} + a_{22}x_{2}+ a_{23}x_{3} = b_{2}$
$a_{31}x_{1} + a_{32}x_{2}+ a_{33}x_{3} = b_{3}$$a_{31}x_{1} + a_{32}x_{2}+ a_{33}x_{3} = b_{3}$

where a’s and b’s are real number constants.

The same equations can be represented using a two-dimensional system called augmented matrix. We usually refer as ‘matrix‘,but the word ‘augmented’ highlights that context of matrix system. We now know from where the matrix came from.

$\begin{bmatrix}a_{11} & a_{12} & ... & a_{1n} & b_{1}\\ a_{21} & a_{22} & ... & a_{2n} & b_{2}\\: & : & : & : & :\\a_{m1} & a_{m2} & ... & a_{mn} & b_{m}\end{bmatrix}$$\begin{bmatrix}a_{11} & a_{12} & ... & a_{1n} & b_{1}\\ a_{21} & a_{22} & ... & a_{2n} & b_{2}\\: & : & : & : & :\\a_{m1} & a_{m2} & ... & a_{mn} & b_{m}\end{bmatrix}$

The double subscript indicate the correct position of a term with respect to rows and columns where rows are equations and columns are terms.

Important points

Here are some important points to consider while constructing a augmented matrix form a system of equations.

• keep the order of a’s according to unknown variables x’s.
• keep the b constant on the right in the same order as in the equation.

Example

Problem #1 : Change the following system of linear equations
to augmented matrix.
-------------------------------------------------------------

$3x_{1} + x_{2}+ 6x_{3} = 1$$3x_{1} + x_{2}+ 6x_{3} = 1$
$2x_{1} + 2x_{2}+ x_{3} = 7$$2x_{1} + 2x_{2}+ x_{3} = 7$
$4x_{1} - 6x_{2}+ 9x_{3} = 5$$4x_{1} - 6x_{2}+ 9x_{3} = 5$

Solution :
------------
The resultant augmented matrix is

$\begin{bmatrix}3 & 1 & 6 & 1\\ 2 & 2 & 1 & 7\\ 4 & -6 & 9 & 5\end{bmatrix}$$\begin{bmatrix}3 & 1 & 6 & 1\\ 2 & 2 & 1 & 7\\ 4 & -6 & 9 & 5\end{bmatrix}$
﻿

We will consider changing the terms with the help of basic row operations in the next post.