What do we gain by representing solutions of Systems of Linear Equations as Parametric Equations? Consider the following system of linear equations:
\begin{aligned}
    4x − 2y &= 1 \\
    16x − 8y &= 4
\end{aligned}
We can eliminate $x$ from the second equation by adding $(−4) \times$ the first
equation to the second. This yields the simplified system
\begin{aligned}
    4x − 2y &= 1 \\
          0 &= 0 \\
\end{aligned}
The second equation does not impose any restrictions on $x$ and $y$ and hence can be omitted. Thus, the solutions of the system are those values of $x$ and $y$ that satisfy the single equation
$$
    4x − 2y = 1
$$
One way to describe the solution set is to solve this equation for $x$ in
terms of $y$ to obtain $x = \frac{1}{4} + \frac{1}{2}y$ and then assign an arbitrary value $t$ (called a parameter) to $y$. This allows us to express the solution by the pair of equations (called parametric equations)
$$
    x = \frac{1}{4} + \frac{1}{2}t, \qquad y = t
$$
My Question is: What do we gain by representing the 2nd-last equation in a parametric way? Are there any obvious advantages that I am not seeing ? I have some understanding of parametric equations but I am unable to relate the solution of linear system to the concept of parametric equations ? or stated otherwise how do they relate to each other ?
 A: The following is more of a philosophical nature. It does not refer to your specific set of equations.
An equation $$\Phi(x)=0,\quad x\in X,\tag{1}$$ which might also be a system of equations, implicitly defines a solution set $$S:=\bigl\{x\in X\bigm|\Phi(x)=0\bigr\} \ \subset X\ .$$ For any trial point $x\in X$ it is easy to check whether it belongs to  $S$ or not, but we don't have a birds eye view over the set $S$. Solving the equation means producing an explicit representation of $S$ in the form of a "list". This list may be empty (if there are no solutions), or consist of exactly one entry, which is then called the solution of $(1)$. As long as $S$ is a finite set we can present it in the form $S=\{a, b,\ldots, p\}$. If the set $S$ is infinite we have to resort to a "production scheme" consisting of an index set $I$ and a function $f:\>I\to X$ that outputs for each $\iota\in I$ an element $x_\iota\in S$ in an essentially bijective way. Such an $f$ is then called a parametric representation of $S$.
In your example we can write $S$ in the form $S=f({\mathbb R})$ with
$$f:\quad{\mathbb R}\to{\mathbb R}^2,\qquad t\mapsto\left({1\over4}+{t\over2},\>t\right)\ .\tag{1}$$
Here $X={\mathbb R}^2$, the index set is the $t$-axis ${\mathbb R}$, and the $f$ in $(1)$is the "production scheme" for $S$.
A: Comparing
$$x(y) = \frac{1}{4} + \frac{1}{2}y$$
vs
$$x(t) = \frac{1}{4} + \frac{1}{2}t, \qquad y(t) = t$$
In the first you represent one of the original variables in terms of the other, describing your solution set with the mapping
$$y \to (x(y),y)$$
In the second it's not much different other than using an arbitrary variable, $ \ t$:
$$t \to (x(t),y(t))=\left(\frac{1}{2}t+\frac{1}{4},t\right)$$
Which only will yield the same set of $(x,y)$'s just like above.  But here you can make the substitution 
$$\quad t\to 2t-\frac{1}{2} \quad $$
for example, leading you to another description of your solution set:
$$t\to (x_2(t),y_2(t))=\left(t,2t-\frac{1}{2}\right)$$
Which again yields that very same set of $(x,y)$'s.  
Using an arbitrary parameter is a degree of freedom that, say, the first solution set description doesn't have.  It gives you a way of describing your solution set in more than one way, which will come into play later in your linear algebraic studies, and in higher dimensions.
A: In your example the solution of the linear system is a line.
Here are some representations of a line in 2D I use:
$$
y = a x + b \\
a x + b y + c = 0 \\
n \cdot (x, y)^\top = d \\
(x, y)^\top = (p_x, p_y)^\top + t\, (d_x, d_y)^\top \\
(1-\lambda) u_1 + \lambda u_2
$$
It depends on the problem, what form I use. E.g. the parametric case is interesting for physical problems, where we are interested in the movement of a point on the line.
If we move to three dimensions, I would need to update that list, e.g. using the intersection of linear equations / affine planes, I would need two of them, and they need to be linear independent:
$$
n_1 \cdot (x, y, z)^\top = d_1 \\
n_2 \cdot (x, y, z)^\top = d_2 \\
$$
But I could keep the parametric forms more or less:
$$
(x, y, z)^\top = (p_x, p_y, p_z)^\top + t \, (d_x, d_y, d_z)^\top \\
(1-\lambda) u_1 + \lambda u_2
$$
