Finite Mathematics, Chapter 1, 1.1, Section 1.1, Problem 34

Determine a equation in slope intercept form (where possible) for the line with $y$-intercept of
$\displaystyle -\frac{2}{3}$ and is perpendicular to $2x - y = 4$

If we transform the given line into point slope form, we have

$
\begin{equation}
\begin{aligned}
2x - y &= 4 \\
\\
y &= 2x - 4
\end{aligned}
\end{equation}
$

Now that the line is in the slope intercept form $y = mx + b$. By observation, $m = 2$
Thus, the slope of the perpendicular line is
$\displaystyle m_{\perp} = -\frac{1}{2}$.
Now, if the line has $x$-intercept of $\displaystyle \frac{-2}{3}$, it means that it passes through
the point $\displaystyle \left( -\frac{2}{3}, 0 \right)$. Then,
By using the point slope form,the equation of the line will be $y - y_1 = m(x - x_1)$

$
\begin{equation}
\begin{aligned}
y - 0 &= - \frac{1}{2} \left( x - \left( -\frac{2}{3} \right) \right)\\
\\
y &= - \frac{1}{2} \left( x + \frac{2}{3} \right)\\
\\
y &= -\frac{1}{2} x - \frac{1}{3}
\end{aligned}
\end{equation}
$