Final answer:
The equation of the line perpendicular to 5x - 3y that contains the point (4,5) is y = -⅓x + ⅙. This was determined by first finding the slope of the original line (5/3), then finding the perpendicular slope (-3/5), and lastly using the point-slope form with the point (4,5) and the perpendicular slope.
Step-by-step explanation:
To find the equation of a line perpendicular to the given line 5x - 3y, we need first to express this line in the slope-intercept form, which is y = mx + b, where 'm' is the slope and 'b' is the y-intercept. The given equation lacks an equals sign and a constant term, but if we assume the absent term is equal to zero, we can rewrite it as -3y = -5x. Dividing by -3, we get y = ⅕x which reveals that the slope of the original line is ⅕.
Perpendicular lines have slopes that are negative reciprocals of each other. Therefore, the slope of the line perpendicular to the original line is -⅓. Using the point-slope form, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line and 'm' is the slope, we can substitute the point (4,5) and the slope -⅓ to get the equation: y - 5 = -⅓(x - 4).
Now, we distribute and simplify to write the equation in slope-intercept form: y = -⅓x + ⅓(4) + 5. Multiplying 4 by -⅓ ends up as -⅖, and adding 5 gives us y = -⅓x + ⅓ + 5, which simplifies to y = -⅓x + ⅙. Therefore, the equation of the line perpendicular to 5x - 3y passing through the point (4, 5) is y = -⅓x + ⅙.