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 $\displaystyle y = \frac{-2}{3} x + 6$.