Final answer:
To find a line perpendicular to y = 5/3x - 2, take the negative reciprocal of the slope (5/3) which is -3/5. Apply the point-slope form with the slope -3/5 and passing through (3, -6) to find the perpendicular line's equation y = -3/5x - 21/5.
Step-by-step explanation:
The given equation of the line is y = 5/3x −2, which has a slope of 5/3. To find an equation of a line that is perpendicular to this one, we can use the negative reciprocal of the original slope, which is −3/5. A line perpendicular to y = 5/3x −2 with a slope of −3/5 that also passes through the point (3, −6) can be found by using the point-slope form of a linear equation which is y − y1 = m(x − x1), where m is the slope and (x1, y1) is the point the line passes through.
Plugging in our values, we get y − (−6) = (−3/5)(x − 3), leading to y + 6 = −3/5x + 9/5. Simplifying further to get the equation in slope-intercept form, we subtract 6 from both sides to obtain the perpendicular line equation, which is y = −3/5x − 21/5.