What is the equation of the line that passes through #(-1,1) # and is perpendicular to the line that passes through the following points: #(13,-1),(8,4) #?

First, we need to find the slope of the for 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)(4) - color(blue)(-1))/(color(red)(8) - color(blue)(13)) = (color(red)(4) + color(blue)(1))/(color(red)(8) - color(blue)(13)) = 5/-5 = -1#

Let's call the slope for the line perpendicular to this #m_p#

The rule of perpendicular slopes is: #m_p = -1/m#

Substituting the slope we calculated gives:

#m_p = (-1)/-1 = 1#

We can now use the point-slope formula to write an equation for the line. The point-slope form of a linear equation is: #(y - color(blue)(y_1)) = color(red)(m)(x - color(blue)(x_1))#

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

Substituting the slope we calculated and the values from the point in the problem gives:

#(y - color(blue)(1)) = color(red)(1)(x - color(blue)(-1))#

#(y - color(blue)(1)) = color(red)(1)(x + color(blue)(1))#

We can also use the slope-intercept formula. 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.

Substituting the slope we calculated gives:

#y = color(red)(1)x + color(blue)(b)#

We can now substitute the values from the point in the problem for #x# and #y# and solve for #color(blue)(b)#

#1 = (color(red)(1) xx -1) + color(blue)(b)#

#1 = -1 + color(blue)(b)#

#color(red)(1) + 1 = color(red)(1) - 1 + color(blue)(b)#

#2 = 0 + color(blue)(b)#

#2 = color(blue)(b)#

Substituting this into the formula with the slope gives:

#y = color(red)(1)x + color(blue)(2)#

The slope of the line passing through #(13,-1) and (8,4)# is

#m_1= (y_2-y_1)/(x_2-x_1)= (4+1)/(8-13)=5/-5 =-1 #

The product of slopes of two perpendicular lines is #m*m_1=-1#

#:. m= -1/m_1 =-1/-1=1# . So the slope of the line passing

through #(-1,1)# is # m=1#.

The equation of the line passing through #(-1,1)# is

#y-y_1=m(x-x_1) = y -1 = 1( x +1) = y-1 =x+1 or x-y = -2 #.

The equation of the line is # x - y = -2 # [Ans]