What is the equation of the line that passes through #(5,7)# and is perpendicular to the line that passes through the following points: #(1,3),(-2,8) #?

First, we will find the slope of the perpendicular line. The slope can be found by using the formula: #m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))#

Where #m# is the slope and (#color(blue)(x_1, y_1)#) and (#color(red)(x_2, y_2)#) are the two points on the line.

Substituting the two points from the problem gives:

#m = (color(red)(8) - color(blue)(3))/(color(red)(-2) - color(blue)(1))#

#m = 5/-3#

A perpendicular line will have a slope (let's call it #m_p#) which is the negative inverse of the line or #m_p = -1/m#

Substituting gives #m_p = - -3/5 = 3/5#

Now that we have the slope of the perpendicular line and one point we can use the point-slope formula to find the equation. The point-slope formula states: #(y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))#

Where #color(blue)(m)# is the slope and #color(red)(((x_1, y_1)))# is a point the line passes through.

Substituting the perpendicular slope we calculated and using the point from the problem gives:

#(y - color(red)(7)) = color(blue)(3/5)(x - color(red)(5))#

Or, if we solve for #y#:

#y - color(red)(7) = (color(blue)(3/5) xx x) - (color(blue)(3/5) xx color(red)(5))#

#y - color(red)(7) = 3/5x - 3#

#y - color(red)(7) + 7 = 3/5x - 3 + 7#

#y - 0 = 3/5x + 4#

#y = 3/5x + 4#