To find the equation of a line in slope-intercept form (which is \( y = mx + b \)), we need the slope \( m \) and the y-intercept \( b \).
We are given:
- A point \((-6, 3)\)
- A slope \( m = \frac{5}{3} \)
Using the slope-intercept form, we can first substitute the slope and the coordinates of the given point into the point-slope formula, which is:
\[ y - y_1 = m(x - x_1) \]
Substituting \( m = \frac{5}{3} \), \( x_1 = -6 \), and \( y_1 = 3 \):
\[ y - 3 = \frac{5}{3}(x + 6) \]
Now, we simplify the equation. First, distribute the slope on the right side:
\[ y - 3 = \frac{5}{3}x + \frac{5}{3} \cdot 6 \]
Calculating \( \frac{5}{3} \cdot 6 \):
\[ \frac{5 \cdot 6}{3} = \frac{30}{3} = 10 \]
So, we have:
\[ y - 3 = \frac{5}{3}x + 10 \]
Next, add 3 to both sides to isolate \( y \):
\[ y = \frac{5}{3}x + 10 + 3 \] \[ y = \frac{5}{3}x + 13 \]
Thus, the equation of the line in slope-intercept form is:
\[ \boxed{y = \frac{5}{3}x + 13} \]