To find the equation of the line passing through the points \((-5, 6)\) and \((5, 2)\), we first need to determine the slope (m) of the line. The slope can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For the points \((-5, 6)\) (where \(x_1 = -5\) and \(y_1 = 6\)) and \((5, 2)\) (where \(x_2 = 5\) and \(y_2 = 2\)), we can substitute these coordinates into the formula:
\[ m = \frac{2 - 6}{5 - (-5)} = \frac{2 - 6}{5 + 5} = \frac{-4}{10} = -\frac{2}{5} \]
Now that we have the slope \(m = -\frac{2}{5}\), we can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
We can use either of the two points; let's use \((-5, 6)\):
\[ y - 6 = -\frac{2}{5}(x - (-5)) \]
This simplifies to:
\[ y - 6 = -\frac{2}{5}(x + 5) \]
Now, distribute the slope:
\[ y - 6 = -\frac{2}{5}x - 2 \]
Now, add 6 to both sides:
\[ y = -\frac{2}{5}x - 2 + 6 \]
This simplifies to:
\[ y = -\frac{2}{5}x + 4 \]
Thus, the equation of the line in slope-intercept form is:
\[ \boxed{y = -\frac{2}{5}x + 4} \]