through: \( (5,2) \), perp. to \( y=-\frac{5}{3} x-2 \)

To find the equation of the line that goes through the point \( (5, 2) \) and is perpendicular to the line given by the equation \( y = -\frac{5}{3} x - 2 \), we first need to determine the slope of the perpendicular line. The slope of the given line is \( -\frac{5}{3} \), so the slope of the perpendicular line will be the negative reciprocal, which is \( \frac{3}{5} \). Using the point-slope form of the equation of a line, we can plug in the point \( (5, 2) \) and the slope \( \frac{3}{5} \): \[ y - 2 = \frac{3}{5}(x - 5) \] Now, simplifying this equation: \[ y - 2 = \frac{3}{5}x - 3 \] Adding 2 to both sides gives: \[ y = \frac{3}{5}x - 1 \] The final equation of the line that passes through \( (5, 2) \) and is perpendicular to \( y = -\frac{5}{3} x - 2 \) is: \[ y = \frac{3}{5}x - 1 \]