Answer
We show that the slope of the line under $G$ is $\frac{{\left( {5 + 3m} \right)}}{{\left( {2 + m} \right)}}$ through the origin in the $xy$-plane.
Work Step by Step
Using $G\left( {u,v} \right) = \left( {2u + v,5u + 3v} \right)$, the image of the line $v = mu$ is
$G\left( {u,mu} \right) = \left( {2u + mu,5u + 3mu} \right)$
So, $x = \left( {2 + m} \right)u$ and $y = \left( {5 + 3m} \right)u$.
Thus, the slope of the line through the origin in the $xy$-plane is given by
$\frac{{y - 0}}{{x - 0}} = \frac{{\left( {5 + 3m} \right)u}}{{\left( {2 + m} \right)u}} = \frac{{\left( {5 + 3m} \right)}}{{\left( {2 + m} \right)}}$
