We are asked to find the equation of a line with slope $m = \frac{3}{5}$ that passes through the point $(5, -2)$.

1. Problem Description

We are asked to find the equation of a line with slope $m = \frac{3}{5}$ that passes through the point $(5, -2)$.

2. Solution Steps

We can use the point-slope form of a line, which is:

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

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

In our case, $m = \frac{3}{5}$ and $(x_1, y_1) = (5, -2)$. Plugging these values into the point-slope form, we get:

$y - (-2) = \frac{3}{5}(x - 5)$

$y + 2 = \frac{3}{5}x - \frac{3}{5}(5)$

$y + 2 = \frac{3}{5}x - 3$

Now, we can solve for $y$ to get the slope-intercept form:

$y = \frac{3}{5}x - 3 - 2$

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

3. Final Answer

The equation of the line is $y = \frac{3}{5}x - 5$.