Let $p(x) = ax + b$ and $q(x) = cx + d$ be two linear polynomials such that:

(A) The graph of $p(x)$ passes through the points $(2, 3)$ and $(6, 11)$.

(B) The graph of $q(x)$ passes through the point $(4, -1)$.

(C) The graph of $q(x)$ is parallel to the graph of $p(x)$.

Find the polynomials $p(x)$ and $q(x)$. Also, find the coordinates of the point where these lines meet the $x$-axis.

(A) For $p(x): 2a + b = 3$ and $6a + b = 11$.

Subtract: $4a = 8 \Rightarrow a = 2$;

then $b = 3 - 4 = -1$.

So, $p(x) = 2x - 1$.

(B) Parallel lines have equal slopes, so $c = 2$.

$q(x)$ passes through $(4, -1)$:

$$-1 = 2(4) + d \Rightarrow d = -9.$$

So, $q(x) = 2x - 9$.

(C) $p(x)$ cuts $x$-axis:

$$2x - 1 = 0$$

$$\Rightarrow x = \frac{1}{2} \Rightarrow \left(\frac{1}{2}, 0\right)$$

$q(x)$ cuts $x$-axis:

$$2x - 9 = 0$$

$$\Rightarrow x = \frac{9}{2} \Rightarrow \left(\frac{9}{2}, 0\right)$$