Answer
$c\left( t \right) = \left( {x\left( t \right),y\left( t \right)} \right) = \left( {2t + 1,4t + 3} \right)$
Work Step by Step
We may write $x\left(t\right)=2t+1$, so that $y\left(t\right)=2\left(2t+1\right)+1=4t+3$. So, the parametrization is
$c\left( t \right) = \left( {x\left( t \right),y\left( t \right)} \right) = \left( {2t + 1,4t + 3} \right)$
When $t=0$, we have $c\left(0\right)=\left({1,3}\right)$. When $t=1$, we have $c\left(1\right)=\left({3,7}\right)$.
Thus, the path $c\left(t\right)$ traces the line $y=2x+1$ from $\left({1,3}\right)$ to $\left( {3,7} \right)$ for $0 \le t \le 1$.
