To find the equation of line \( v \), we first need to determine the slope of line \( u \) from its equation.
The equation of line \( u \) is given in point-slope form: \[ y - 5 = 2(x - 5) \] From this equation, we can see that the slope \( m \) of line \( u \) is \( 2 \).
Since line \( v \) is parallel to line \( u \), it will have the same slope. Therefore, the slope of line \( v \) is also \( 2 \).
Next, we can use the point-slope form of the equation of a line to find the equation of line \( v \). Line \( v \) passes through the point \( (-2, -3) \) and has a slope of \( 2 \). The point-slope form is given by:
\[ y - y_1 = m(x - x_1) \] Plugging in \( m = 2 \), \( x_1 = -2 \), and \( y_1 = -3 \): \[ y - (-3) = 2(x - (-2)) \] This simplifies to: \[ y + 3 = 2(x + 2) \]
Now, distribute the \( 2 \): \[ y + 3 = 2x + 4 \]
Next, we subtract \( 3 \) from both sides to put the equation in slope-intercept form \( y = mx + b \): \[ y = 2x + 4 - 3 \] \[ y = 2x + 1 \]
Thus, the equation of line \( v \) in slope-intercept form is: \[ \boxed{y = 2x + 1} \]