How do you determine the equation of a line passing through (2, -3) that is perpendicular to the #4x-y=22#?

1 Answer
Dec 18, 2016

Answer:

#y + 3 = -1/4x + 1/2#

or

#y = -1/4x - 5/2#

Explanation:

To determine this perpendicular line we can use the point-slope formula. We already have a point (2, -3), now we need to determine the slope. The slope of a perpendicular line is the negative inverse of the line it is perpendicular to. If we convert the equation we are given to the slope-intercept form we will have a slope we can take the negative inverse of.

#4x - y = 22#

#4x - y color(red)( + y - 22) = 22color(red)( + y - 22)#

#4x - 22 = y#

#y = 4x - 22#

The slope-intercept form is #color(red)(y = mx + b)# where #color(red)(m)# is the slope. Therefore the slope of the line we were give is #m = 4#

Given a slope #color(red)(m)# the negative inverse is #color(red)(-1/m)#

For our slope of #m = 4# the negative inverse is #-1/4#

Now we can use this slope and the point we were given and use the point-slope formula to find the equation of the perpendicular line.

The point-slope formula states: #color(red)((y - y_1) = m(x - x_1))#
Where #color(red)(m)# is the slope and #color(red)((x_1, y_1))# is a point the line passes through.

Substituting the information we have gives:

#y - -3 = -1/4(x - 2)#

#y + 3 = -1/4x + 2/4#

#y + 3 = -1/4x + 1/2#

If we want to convert to the more standard slope-intercept form we would get:

#y + 3 color(red)( - 3) = -1/4x + 1/2 color(red)( - 3)#

#y = -1/4x + 1/2 - 6/2#

#y = -1/4x - 5/2#