#### Answer

$a = \frac{1}{6}$

#### Work Step by Step

This equation is in slope-intercept form, which is given by the formula:
$y = mx + b$,
where $m$ is the slope of the line and $b$ is the y-intercept of the line.
If we look at the equation we are given, $y = -4ax - 10$, then $-4a$ would be $m$, our slope. We know that the actual value for our slope is $m = -\frac{2}{3}$, so we set $-4a$ equal to $-\frac{2}{3}$, and then solve for $a$:
$-4a = -\frac{2}{3}$
To solve for $a$, we divide both sides of the equation by $-4$, which means that we multiply by the reciprocal, which is $-\frac{1}{4}$:
$(-\frac{1}{4})(-4)a = (-\frac{1}{4})(-\frac{2}{3})$
Multiply to simplify:
$a = \frac{2}{12}$
Simplify the fraction by dividing both the numerator and denominator by their greatest common factor, which is $2$:
$a = \frac{1}{6}$