What is the slope of any line perpendicular to the line passing through #(7,23)# and #(1,2)#?

See the entires solution process below.

Explanation:

First, we need to determine the slope of the line passing through the two points. 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 values from the points in the problem gives:

#m = (color(red)(2) - color(blue)(23))/(color(red)(1) - color(blue)(7)) = (-21)/-6 = (-3 xx 7)/(-3 xx 2) = (color(red)(cancel(color(black)(-3))) xx 7)/(color(red)(cancel(color(black)(-3))) xx 2) = 7/2#

So the slope of any line perpendicular to this line, let's call this slope #m_p#, will be the negative inverse of the slope of the line it is perpendicular to, or:

#m_p = -1/m#

Therefore, for the problem:

#m_p = -2/7#