What is an equation in slope-intercept form of the line that is perpendicular to the graph of # y= -4x+5# and passes through (1, 1)?

Slope of the line, # y= -4 x +5 or ; [y=mx+c]#

is #m_1= -4# [Compared with slope-intercept form of equation]

The product of slopes of the pependicular lines is #m_1*m_2=-1#

#:.m_2=(-1)/-4=1/4#. The equation of line passing through

#(x_1=1,y_1=1)# having slope of #m_2# is #y-y_1=m_2(x-x_1)#.

#:. y-1=1/4(x-1) or y = 1/4 x -1/4+1 or y= 1/4 x +3/4 #.

Equation of line in slope-intercept form is #y=1/4 x +3/4# [Ans]

The given equation #y=color(green)(-4)x+color(red)5#
is in slope-vertex form with
slope #color(green)m=color(green)(-4)#, and
y-intercept #color(red)b=color(red)5#

If a line has a slop of #color(green)m#
then every line perpendicular to it has a slope of #color(blue)(-1/m)#.

Therefore all lines perpendicular to #y=color(green)(-4)+5#
will have a slope of #color(blue)(1/4)#
and will have an equation in slope-intercept form:
#color(white)("XXX")y=color(blue)(1/4)x+color(magenta)c# for some constant (the y-intercept) #color(magenta)c#

If #(color(brown)x,color(lime)y)=(color(brown)1,color(lime)1)# is a solution for the required line with this form,
then
#color(white)("XXX")color(lime)1=color(blue)(1/4) * color(brown)1 + color(magenta)c#

#color(white)("XXX")rArr color(magenta)c=3/4#

Therefore the resolved equation for the given line is
#color(white)("XXX")y=1/4x+3/4#