Answer
We show that the image of the horizontal line $v=c$ is the line $y = \frac{5}{2}x + \frac{1}{2}c$.
The image of the vertical line $u=c$ is the line $y = 3x - c$.
Work Step by Step
Using the mapping $G\left( {u,v} \right) = \left( {2u + v,5u + 3v} \right)$, the image of the horizontal line $v=c$ is given by
$G\left( {u,c} \right) = \left( {2u + c,5u + 3c} \right)$
So, $x=2u+c$ and $y=5u+3c$.
Substituting $u = \frac{{x - c}}{2}$ in $y=5u+3c$ gives
$y = 5\left( {\frac{{x - c}}{2}} \right) + 3c$
$y = \frac{5}{2}x + \frac{1}{2}c$
Hence, the image of the horizontal line $v=c$ is the line $y = \frac{5}{2}x + \frac{1}{2}c$.
Using the mapping $G\left( {u,v} \right) = \left( {2u + v,5u + 3v} \right)$, the image of the vertical line $u=c$ is given by
$G\left( {c,v} \right) = \left( {2c + v,5c + 3v} \right)$
So, $x=2c+v$ and $y=5c+3v$
Substituting $v=x-2c$ in $y=5c+3v$ gives
$y = 5c + 3\left( {x - 2c} \right)$
$y = 3x - c$
Thus, the image of the vertical line $u=c$ is the line $y = 3x - c$.
