A horizontal line has infinitely many values for $\displaystyle x$, but only one possible value for $\displaystyle y$. Thus, it is always of the form $\displaystyle y = c$, where $\displaystyle c$ is a constant. Horizontal lines have a slope of $\displaystyle 0$. The only equation of this form is $\displaystyle y = 0$. 

The equation $\displaystyle x=5$ is a vertical line, so the perpendicular line must be horizontal. The only answer choice that is a horizontal line is $\displaystyle y = -5$.

A vertical line has infinitely many values of $\displaystyle y$ but only one value of $\displaystyle x$. Thus, vertical lines are of the form $\displaystyle x = c$, where $\displaystyle c$ is a real number. The only equation of this form is $\displaystyle x = 2$. 

The graph of x=5 is a vertical line. The equation x=5 represents all points with x- value equal to 5.

Try to plot a couple of points with an x-value of 5.

A few examples are (5, 0), (5, 2), (5,5).

Draw a line connecting the points and you obtain a vertical line intercepting the x-axis at (5,0).

Think about the meaning of a vertical line on the coordinate grid. The $\displaystyle y$ value changes to any value, yet the $\displaystyle x$ value always stays the same. Thus, we are talking about an equation in which the $\displaystyle y$ is free, or is not effected, and the $\displaystyle x$ is constant. This is an equation of the form $\displaystyle x = c$, where $\displaystyle c$ is a constant. 

Think about what it means to be a horizontal line. The $\displaystyle x$ value changes to be any real number, but the $\displaystyle y$ value always remains constant. Thus, we are looking for an equation in which the $\displaystyle y$ value is constant and the  $\displaystyle x$ value is not present. This would be any equation of the form $\displaystyle y = c$, where $\displaystyle c$ is a constant. 

A vertical line is only dependent on $\displaystyle x$ and is completely independent of $\displaystyle y$. Therefore, since I is $\displaystyle x=-4$, I is a vertical line. II is not a vertical line since it is only dependent upon $\displaystyle y$; it is in fact a horizontal line.

For III, if we simplify the equation

$\displaystyle \underset{-3y}{3y}-5=\underset{-3y}{3y}+4x+7$

it becomes

$\displaystyle -\underset{-7}{5}=4x+\underset{-7}{7}$

and finally

$\displaystyle \frac{4x}{4}=\frac{-12}{4}=-3=x$

Since III is only dependent upon $\displaystyle x$, III is a vertical line.

IV is codependent upon both $\displaystyle x$ and $\displaystyle y$, so it is neither a vertical nor a horizontal line. Therefore, only I and III are vertical lines.

The equation for a horizontal line is $\displaystyle y=b$

where b is the y-coordinate of the point $\displaystyle (a, b)$ on the line.

As such, the equation for the line containing the point (7, 10) is

$\displaystyle y = 10$

The equation for a vertical line is $\displaystyle x=a$ where $\displaystyle a$ is the $\displaystyle x$-coordinate of the point $\displaystyle (a, b)$ on the line.

As such, the equation for the line containing the point $\displaystyle (3, 7)$ is,

$\displaystyle x = 3$.

The slope of a line is defined as the change in the y direction divided by the change in the x direction. A vertical line does not change in the y direction nor the x direction. This means that it's slope is $\displaystyle \frac{0}{0}$. It is impossible to divide any number by zero, therefore the answer is undefined.