Step 1

given simultaneous equations are,

$ax+by+c=0$ and

$dx+ey+f=0$

Step 2

given system can be written as,

$ax+by=-c$ and

$dx+ey\pm f$

above system is non homogeneous system of equation,

compaire it with $ax=b$

$\left[\begin{array}{cc}a& b\\ d& e\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}-c\\ -f\end{array}\right]$

when $ae=bd$ then we get a solution in integer.

and if $ae\ne bd$ then we have to solve system of simultaneous equation by using row transformation.

$\left[\begin{array}{cc}a& b\\ d& e\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}-c\\ -f\end{array}\right]$

$\frac{1}{a}{R}_{1}\Rightarrow \left[\begin{array}{cc}1& \frac{b}{a}\\ d& e\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}-\frac{c}{a}\\ -f\end{array}\right]$

$R}_{2}={R}_{2}-d{R}_{1$ gives,

$\left[\begin{array}{cc}1& \frac{b}{a}\\ 0& e-(\frac{bd}{a})\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}-\frac{c}{a}\\ -f+cd\end{array}\right]$

by solving x and y we get rational expressions,

so we have a rational expressions in $\{a,b,c,d,e,f\}$ whenever $ae\ne bd$.