The question requires finding two linear polynomials $p(x) = ax + b$ and $q(x) = cx + d$ using three given conditions.

From condition (i), we find $b = 5$ since $p(0) = a(0) + b = 5$.

From condition (iii), equating coefficients of $p(x) + q(x) = 6x + 4$ gives $a + c = 6$ and $b + d = 4$, allowing us to find $d = -1$.

From condition (ii), since $p(x) - q(x)$ cuts the x-axis at $(3,0)$, substituting $x = 3$ gives $3(a - c) + (b - d) = 0$, yielding $a - c = -2$.

Solving the system of equations $a + c = 6$ and $a - c = -2$ gives $a = 2$ and $c = 4$.