Final answer:
The equation of the line that is perpendicular to 4x - 5y = 12 and passes through the point (20, 12) is y = -5/4x + 37. This is derived by first finding the negative reciprocal of the original line's slope and then using the point-slope form with the given point.
Step-by-step explanation:
To find the equation of a line that is perpendicular to another and passes through a specific point, we first need the slope of the original line. The given equation, 4x - 5y = 12, can be written in slope-intercept form (y = mx + b) as y = (4/5)x - 12/5. The slope of this line is 4/5. Perpendicular lines have slopes that are negative reciprocals of each other. Therefore, the slope of the perpendicular line will be -5/4.
Now, we use the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the point the line passes through, and m is the slope of the line. Substituting the given point (20, 12) and the slope -5/4, we get: y - 12 = -5/4(x - 20).
To put this into slope-intercept form, we distribute the slope on the right side and add 12 to both sides: y = -5/4x + 5(20/4) + 12. Simplifying, we get y = -5/4x + 25 + 12. Combining like terms, the final equation is y = -5/4x + 37. This is the equation of the line perpendicular to 4x - 5y = 12 and passes through (20, 12).