Beginning Algebra With Applications, Chapter 5, 5.4, Section 5.4, Problem 34

a. Use the point-slope formula to find the equation of the line that goes through the point $(5,3)$ and has slope .


$
\begin{equation}
\begin{aligned}

(x_1, y_1) =& (5,3)
&& \text{Let $(x_1, y_1)$ be the given point}
\\
m =& 0
&& \text{$m$ is the given slope}
\\
y - y_1 =& m(x- x_1)
&& \text{Point slope formula}
\\
y -3 =& 0(x-5)
&& \text{Substitute $5$ for $x$, $3$ for $y$ and } 0 \text{ for $m$}
\\
y -3 =& 0
&& \text{Rewrite the equation in the form } y = mx+b
\\
y =& 3
&& \text{Add } 3

\end{aligned}
\end{equation}
$



b. Does your answer to part a simplify to the equation of a horizontal line through $(0,b)$?

Yes because if $m$ is equal to zero in the point slope form... then the expression written in the right side of the equation will be zero as well. What is left is the $y$ coordinate of the point that makes the equation a horizontal line.