#### Answer

(a): $c= \frac{-1}{4}$
(b): $c=-2$
(c): no value of $c$
(d): $c=0$

#### Work Step by Step

(a): $slope = \frac{-1}{c} = 4$, which implies that $c= \frac{-1}{4}$.
(b): The equation $x+cy=1$ must satisfy that the line that passes through $(3,1)$; that is, $3+c=1 \to c=-2$.
(c): In order for the line to be horizontal, we must have the slope $\frac{-1}{c} = 0$, but solving for $c$ would require division by zero, which is not allowed. Thus, there is no value of $c$ in this case.
(d): For the line to be vertical means that the slope $\frac{-1}{c} \to \infty$ or $\frac{-1}{c} \to - \infty$, which means that the value of $c$ must be $0$ in this case.