Final answer:
The equation of the line that passes through the point (5,-2) and is perpendicular to y = \frac{5}{6}x - 5 is y = -\frac{6}{5}x + 12.
Step-by-step explanation:
To find the equation of the line that passes through the point (5,-2) and is perpendicularto y = \frac{5}{6}x - 5, we first determine the slope of the given line. The slope m of the given line is \frac{5}{6}. Two lines that are perpendicular to each other have slopes that are negative reciprocals of each other. Therefore, the slope of the line we are looking for is -\frac{6}{5}.
Now that we have the slope of the desired line, we can use the point-slope form y - y1 = m(x - x1), which yields:
y + 2 = -\frac{6}{5}(x - 5).
We then distribute the slope and move the constant to the other side of the equation to get the slope-intercept form, which gives us:
y = -\frac{6}{5}x + 10 + 2
Finally, combine the constants to obtain the equation in slope-intercept form:
y = -\frac{6}{5}x + 12