The equation of a line is \( y=5 x+2 \). Find the oquation of a line perpendiculat to it and passing through \( \left(4,-\frac{3)}{3}\right. \)

To find the equation of a line perpendicular to \( y = 5x + 2 \), we first need to know that the slope of the given line is \( 5 \). The slope of a line perpendicular to another is the negative reciprocal of the original slope. Hence, the perpendicular slope will be \( -\frac{1}{5} \). Now, using the slope-point form \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point through which the line passes, we plug in the slope \( -\frac{1}{5} \) and the point \( \left(4, -\frac{3}{3}\right) \) or \( (4, -1) \): \[ y - (-1) = -\frac{1}{5}(x - 4) \] This simplifies to: \[ y + 1 = -\frac{1}{5}x + \frac{4}{5} \] Rearranging gives the equation of the perpendicular line: \[ y = -\frac{1}{5}x + \frac{4}{5} - 1 \] \[ y = -\frac{1}{5}x - \frac{1}{5} \] So, the equation is: \[ y = -\frac{1}{5}x - \frac{1}{5} \]