How to show that the linear programme is unbounded objective function? What is the way I can prove any linear program has an unbounded solution? 
Take for example below, its clear that it has infinitely many solutions. However, I want to know a method which tells systematic way of computing if any linear program has an unbounded solution. 

 A: Let's work with a linear program in the form you have used for your example, that is, a problem of the form 
$$
\begin{eqnarray*}
\max & c^Tx \\
\textrm{subject to} & \quad Ax = b\\ 
& x \geq 0
\end{eqnarray*}
$$
You can think of certain vectors $y$ as directions leading to "unboundedness" if they satisfy the following two conditions:


*

*$Ay = 0,\, y \geq 0$,

*$c^Ty > 0$.


The intuition is that any vector $y$ that satisfies condition (1) does not affect feasibility. More concretely, consider any feasible point $x^*$ (which means that $x^* \geq 0$ and $Ax^*=b$). Now consider any number $t > 0$ and the vector obtained by considering the sum $x^* + ty$. Since $x^* \geq 0$, $t > 0$ and $y \geq 0$ it follow that $x^* + ty \geq 0$ (the sum of non-negative quantities is non-negative). Also notice that
$$
A(x^* + ty) = Ax^* + A(ty) = Ax^* + t(Ay) = b + t(0) = b.
$$ 
So it turns out that $x^* + ty$ is a feasible solution for your linear program. Moreover, this holds for any $t > 0$. If you trace out all the values of $(x^* + ty)$ for all $t > 0$ you end up with a ray with base point $x^*$ and direction $y$. Think of this ray as a "half-line" of infinite length in one direction starting at the point $x^*$.
Now let's consider condition (2) which tells you what happens when you scale $y$. Suppose that $c^Ty = v$; condition (2) tells us that $v > 0$. Choose any $t > 1$ and observe that
$$
c^T(t\cdot y) = t \cdot c^Ty = tv > v.
$$
If you look more closely at the steps above you will see that $c^Ty$ keeps increasing proportionally as we increase $t$ which means that $c^Ty \to \infty$ as $t \to \infty$.
So a linear program is unbounded if it has a feasible solution (our $x^*$ above) and a vector $y$ that satisfies conditions (1) and (2). This provides a systematic way to show that a linear program is unbounded.
