To find the equation of line $q$, we need to follow these steps:

Step 1: Identify the slope of line $p$.

The equation of line $p$ is given as:

$y = \frac{5}{3}x - 4$

This is in slope-intercept form $y = mx + b$, where $m$ is the slope. For line $p$, the slope $m_p = \frac{5}{3}$.

Step 2: Find the slope of line $q$.

Since line $q$ is perpendicular to line $p$, its slope $m_q$ will be the negative reciprocal of the slope of line $p$.

The slope of line $p$ is $\frac{5}{3}$, so the slope of line $q$ will be:

$m_q = -\frac{3}{5}$

Step 3: Use the point-slope form to write the equation of line $q$.

We are given that line $q$ passes through the point $(-10, -3)$. The point-slope form of the equation of a line is:

$y - y_1 = m(x - x_1)$

where $(x_1, y_1)$ is a point on the line, and $m$ is the slope.

Substituting the slope $m_q = -\frac{3}{5}$ and the point $(-10, -3)$, we get:

$y - (-3) = -\frac{3}{5}(x - (-10))$

Simplifying:

$y + 3 = -\frac{3}{5}(x + 10)$

Step 4: Simplify the equation to slope-intercept form.

Distribute the slope $-\frac{3}{5}$ on the right side:

$y + 3 = -\frac{3}{5}x - 6$

Now subtract 3 from both sides to solve for $y$:

$y = -\frac{3}{5}x - 6 - 3$

$y = -\frac{3}{5}x - 9$

Thus, the equation of line $q$ is:

$y = -\frac{3}{5}x - 9$

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Do you need further details or clarification on any step?

Here are five related questions to explore further:

1. What does the negative reciprocal property of perpendicular lines mean geometrically?
2. How can we convert from point-slope form to standard form for linear equations?
3. How can we graph the two lines $p$ and $q$ on a coordinate plane?
4. What would happen if the lines were parallel instead of perpendicular?
5. How do we find the equation of a line given two points instead of a point and a slope?

Tip: To find the slope of a line given its equation in standard form $Ax + By = C$, the slope is $-\frac{A}{B}$.