Answer
The different sets of parametric equation are as follows:
$ x=t\text{, }\,y=4t-3\,\,\text{ and }\,\,x=t+1\text{,}\,\,y=4t+1$
Work Step by Step
Choose x values such that solution set for y doesn’t change.
Let us assume $ x=t $, then $ y=4t-3$
Here, for $ x=t $
And the solution set for y remains the same for both negative and positive values.
Again, let $ x=\,{{t}^{2}}$; then $ y=4{{t}^{2}}-3$
The solution set for y is positive for both negative and positive values of t.
Therefore, $ x=\,{{t}^{2}}$
It cannot be used to determine the set of parametric equations.
Again, Let $ x=\,t+1$, then
$\begin{align}
& y=4\left( t+1 \right)-3 \\
& y=4t+4-3 \\
& y=4t+1 \\
\end{align}$
The solution set for y remains the same for both negative and positive values.
Note that many different sets of parametric equations are possible. Choose any two from them.
Thus, $ x=\,t $, $ y=4t-3$ and $ x=\,t+1$, $ y=4t+1$ are two different sets of parametric equations.
