To find the equation of line \( w \) that is perpendicular to line \( v \) and passes through the point \( (9, 6) \), we first need to determine the slope of line \( v \) from its equation.
The equation of line \( v \) is given as:
\[ y = -\frac{9}{7}x - 5 \]
From this, we can see that the slope \( m_v \) of line \( v \) is \( -\frac{9}{7} \).
Since line \( w \) is perpendicular to line \( v \), its slope \( m_w \) will be the negative reciprocal of \( m_v \):
\[ m_w = -\frac{1}{m_v} = -\frac{1}{-\frac{9}{7}} = \frac{7}{9} \]
Now we have the slope of line \( w \) as \( \frac{7}{9} \). Next, we can use the point-slope form of a line's equation to find the equation of line \( w \) that passes through the point \( (9, 6) \):
The point-slope form is given by:
\[ y - y_1 = m(x - x_1) \]
Where \( (x_1, y_1) \) is a point on the line. Plugging in our point \( (9, 6) \) and the slope \( \frac{7}{9} \):
\[ y - 6 = \frac{7}{9}(x - 9) \]
Now, we will simplify this equation to get it into slope-intercept form \( y = mx + b \).
Expanding the right-hand side:
\[ y - 6 = \frac{7}{9}x - \frac{7}{9} \times 9 \] \[ y - 6 = \frac{7}{9}x - 7 \]
Now, add 6 to both sides:
\[ y = \frac{7}{9}x - 7 + 6 \] \[ y = \frac{7}{9}x - 1 \]
Thus, the equation of line \( w \) in slope-intercept form is:
\[ \boxed{y = \frac{7}{9}x - 1} \]