Given the linear equation 2x + 3y − 8 = 0,

(i) For intersecting lines,

Equation of another intersecting line to the given line is−

$2 x+5 y-3=0$

Since, condition for intersecting lines and unique solution is−

$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$

(ii) For parallel lines,

Equation of another parallel line to the given line is−

$2 x+3 y-3=0$

Since, condition for parallel lines and no solution is−

$\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$

(iii) For co−incident lines,

Equation of another coincident line to the given line is−

$4 x+6 y-16=0$

Since, condition for coincident lines and infinite solution is−

$\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$