Let's solve the problem step by step:

We are given the equation for line $k$:

$y + 5 = \frac{7}{5}(x - 10)$

First, we can rewrite it in slope-intercept form, $y = mx + b$, where $m$ is the slope.

1. Distribute $\frac{7}{5}$ on the right-hand side:

$y + 5 = \frac{7}{5}x - \frac{7}{5} \cdot 10$

$y + 5 = \frac{7}{5}x - 14$

2. Subtract 5 from both sides to isolate $y$:

$y = \frac{7}{5}x - 14 - 5$

$y = \frac{7}{5}x - 19$

So, the slope of line $k$ is $\frac{7}{5}$.

Finding the equation of line $l$

Since line $l$ is parallel to line $k$, it has the same slope $m = \frac{7}{5}$. The equation of line $l$ passes through the point $(-3, -6)$.

Using the point-slope form of the line equation:

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

where $(x_1, y_1) = (-3, -6)$ and $m = \frac{7}{5}$, we can plug in the values:

$y - (-6) = \frac{7}{5}(x - (-3))$

$y + 6 = \frac{7}{5}(x + 3)$

Now distribute $\frac{7}{5}$:

$y + 6 = \frac{7}{5}x + \frac{7}{5} \cdot 3$

$y + 6 = \frac{7}{5}x + \frac{21}{5}$

Subtract 6 from both sides to isolate $y$:

$y = \frac{7}{5}x + \frac{21}{5} - 6$

Convert 6 to a fraction with a denominator of 5:

$y = \frac{7}{5}x + \frac{21}{5} - \frac{30}{5}$

Simplify:

$y = \frac{7}{5}x - \frac{9}{5}$

Final Equation:

The equation of line $l$ in slope-intercept form is:

$y = \frac{7}{5}x - \frac{9}{5}$

Let me know if you want further details or have any questions!

Here are 5 related questions:

1. How do you determine the slope of a line from its equation?
2. What is the point-slope form of a line, and when do you use it?
3. How do you know two lines are parallel?
4. How can you convert a point-slope equation to slope-intercept form?
5. How does changing the slope of a line affect its direction?

Tip: Always check if two lines are parallel by comparing their slopes!