First, we need to find the slope of the line passing through the two points in the problem. 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)(0) - color(blue)(7))/(color(red)(-2) - color(blue)(-8)) = (color(red)(0) - color(blue)(7))/(color(red)(-2) + color(blue)(8)) = -7/6#
The formula for a perpendicular slope #(m_p)# is:
#m_p = -1/m#
Substituting the slope we calculated gives:
#m_p = -1/(-7/6) => 6/7#
We can now use the point-slope formula to write an equation for the line from the problem. 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 the values from the point in the problem gives:
#(y - color(red)(8)) = color(blue)(6/7)(x - color(red)(5))#
If necessary, we can solve for #y# to transform this equation to slope-intercept form. The slope-intercept form of a linear equation is: #y = color(red)(m)x + color(blue)(b)#
Where #color(red)(m)# is the slope and #color(blue)(b)# is the y-intercept value.
#y - color(red)(8) = (color(blue)(6/7) xx x) - (color(blue)(6/7) xx color(red)(5))#
#y - color(red)(8) = 6/7x - 30/7#
#y - color(red)(8) + 8 = 6/7x - 30/7 + 8#
#y - 0 = 6/7x - 30/7 + (7/7 xx 8)#
#y = 6/7x - 30/7 + 56/7#
#y = 6/7x + (-30 + 56)/7#
#y = color(red)(6/7)x + color(blue)(26/7)#
