College Algebra, Chapter 2, 2.4, Section 2.4, Problem 32

Find an equation of the line that has $y$ intercept $6$ and parallel to the line $2x + 3y + 4 = 0$


$
\begin{equation}
\begin{aligned}

y =& mx + b
&& \text{Point Slope Form}
\\
\\
y =& mx + 6
&& \text{Substitute the $y$ intercept}

\end{aligned}
\end{equation}
$



If the line is parallel with $2x + 3y + 4 = 0$, then we can say that they have the same slope.


$
\begin{equation}
\begin{aligned}

2x + 3y + 4 =& 0
&&
\\
\\
3y =& -2x - 4
&& \text{Subtract $2x$ and $4$}
\\
\\
y =& - \frac{2}{3}x - \frac{4}{3}
&& \text{Divide both sides by 3}


\end{aligned}
\end{equation}
$


By observation, the slope of the line is $\displaystyle \frac{-2}{3}$. Thus, the equation of the line is $\displaystyle y = \frac{-2}{3} x + 6$.