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.

```
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.

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### 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

```
```

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.

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

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