How do you find the slope that is perpendicular to the line #4x+5y= -5#?

2 Answers
Jun 7, 2017

Answer:

The slope of the line perpendicular to the line #4x+5y=-5# is #5/4#.

Explanation:

Let's start with the original equation:

#4x+5y=-5#

From here, we can manipulate the equation into the slope-intercept form. We first move the #4x# over to the right side by subtracting #4x# from both sides:

#5y=-5-4x#

Next, we divide both sides by #5# to isolate the #y# variable:

#y=(-5-4x)/5#

We then simplify the right portion of the equation:

#y=-5/5-(4x)/5#

And further simplification follows:

#y=-1-(4x)/5#

We then rearrange the entire equation to clearly show the equation in slope-intercept form:

#y=-(4x)/5-1#

Now that we have the equation in slope-intercept form, we can clearly see that #-4/5# is the slope here.

From here, it is easy. The product of a slope and its perpendicular slope is always #-1#. (Proof: perpendicular lines have negative reciprocal slope)

If we set #a# to be the perpendicular slope of #-4/5#, then

#-4/5a=-1#

Then, we can isolate #a# by dividing by #-4/5# on both sides and then simplifying the result:

#a=(-1)/(-4/5)=-1*-5/4=5/4#

Thus, the slope of the line perpendicular to the line #4x+5y=-5# is #5/4#.

Jun 7, 2017

Answer:

The slope that is perpendicular to the line #4x + 5y = -5# is #5/4#.

Explanation:

The slope perpendicular to a line is the opposite reciprocal of the slope of the given line.

For example, if the slope of a line is 2, the slope of a line perpendicular to the line with slope 2 must be #-1/2#. "Opposite" refers to the opposite sign (e.x. the opposite of #-5# is #5#), and "reciprocal" just means 1 over whatever you're given (e.x. the reciprocal of 20 is just #1/20#).

Because in your case the line is in standard form (i.e. #ax + by = c#, such that #a# is a positive integer, and #b# and #c# are integers), the slope of the line is given by the equation: #m=-a/b#. Thus, the slope of the line is #-4/5#.

You can also find the slope, #m#, by putting the equation into slope-intercept form (#y=mx+b#):
1. #4x + 5y = -5#
2. #5y = -4x - 5#
3. #y = -4/5x -1#
4. #m=-4/5#

The slope of a line perpendicular to this line must be the opposite reciprocal, so #-(1/(-4/5)) = 5/4#. The slope that is perpendicular to the line #4x + 5y = -5# is #5/4#.