How do you write an equation of the line given (1, 5) that are parallel to and perpendicular to the line equation y + 4x = 7?

Put the equation into slope-intercept form

The slope intercept form is
#y = mx + b#
This form of the equation lets you read the slope of the line directly from the equation.

#y + 4x = 7#    Solve for #y#

Subtract #4x# from both sides to isolate #y#
#y = - 4x + 7#

This form of the equation tells you that
the slope of the given line is #-4#

Find the equation of the line parallel to this one, going through the point #(1,5)#

1) All lines parallel to each other have the same slopes.
So the slope of the line parallel to the given line is also #-4#

That means that the parallel line's equation so far is
#y = -4x + b#

2) Now find #b#

Sub in the values for #x# and #y# from the given ordered pair and solve for #b#

#y = -  4   x   + b#
#5 = -(4)(1)+ b#

#5 = -4 + b#
#9 = b#

So the entire equation for the line parallel to the given line is
#y = -4x + 9# #larr# answer

#color(white)(mmmmmmmm)#―――――――――

Find the equation of the line perpendicular to the given line

Lines that are perpendicular to each other have slopes that are the "negative inverse" of each other.

To write the slope of a line perpendicular to the given line, invert the given slope and change its sign.

In this case, that means that the slope of all the lines perpendicular to the given line have the slope #+(1)/(4)#

So the equation of the perpendicular line so far is

#y = (1)/(4)(x) + b#

Solve for #b#

Sub in the values for #x# and #y# from the given ordered pair

#5 = (1)/(4)(1) + b#

Clear the fraction by multiplying all the terms on both sides by #4# and letting the denominator cancel
#20 =1 + 4b#

Subtract #1# from both sides to isolate the #4b# term
#19 = 4b#

Divide both sides by #4# to isolate #b#

#(19)/(4) = b#

Now you can write the entire equation for the line perpendicular to the given line and going through (1,5)

#y = (1)/(4)(x) + (19)/(4)# #larr# answer