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.

\begin{aligned} &a_{1}x_{1} + a_{2}x_{2} = b_{1}\\ &a_{3}x_{1} + a_{4}x_{2} = b_{2} \end{aligned}

where either one variable can be zero, not both.

Let us call the lines as and . A solution to the equation is a point that satisfy both equations given above.

### Possibilities of Solution

There are three possibilities of solution for lines – and .

- No Solutions
- Exactly One Solution
- Infinite Solutions

Let us examine each one of these possibilities closely.

#### No Solutions

If the lines – and are parallel, that is, not touching each other, there is no possibility of a common point .

Therefore, the above system has **no solution.**

#### Exactly One Solution

Suppose the lines – and intersect each other exactly at a point . Then we have exactly one solution to the linear system.

The point in the above diagram is the only solution that satisfy equation of both the lines.

#### Infinite Solutions

When the two lines – and are the same or overlap each other then we can say that there could be more than one points 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 equations and variable , …, is written as

\begin{aligned} &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_{31}x_{1} + a_{32}x_{2}+ a_{33}x_{3} = b_{3} \end{aligned}

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}

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.

\begin{aligned} &3x_{1} + x_{2}+ 6x_{3} = 1\\ &2x_{1} + 2x_{2}+ x_{3} = 7\\ &4x_{1} - 6x_{2}+ 9x_{3} = 5 \end{aligned}

Solution : The resultant augmented matrix is

\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.