What is an equation in slope-intercept form of the line that is perpendicular to the graph of #y=2x+3# and passes through (3, -4)?

Question

Avery Allen · Accepted Answer

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

The slope of the line perpendicular to the graph of #y=2x+3# is #-1/2#. The perpendicular slope is the negative inverse of the original slope. The product of perpendicular slopes is #-1#, where:

#m_1m_2=-1#,

where:

#m_1# is the original slope #(2)# and #m_2# is the perpendicular slope.

#2m_2=-1#

Divide both sides by #2#.

#m_2=-1/2#

So we now have the slope and we have been given a point #(color(red)3,color(blue)(-4))#.

Find the point-slope form of the perpendicular line.

#y-y_1=m(x-x_1)#

Plug in the known values.

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

#y+4=-1/2(x-3)# #larr# point-slope form.

To convert the point-slope form to slope-intercept form, solve the point-slope form for #y#.

Slope-intercept form is: #y=mx+b#, where #m# is the slope and #b# is the y-intercept.

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

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

Subtract #4# from both sides.j

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

Multiply #4# by #2/2# to get an equivalent fraction with #2# as the denominator.

#y=-1/2x+3/2-4xx2/2#

Simplify.

#y=-1/2x+3/2-8/2#

Simplify.

#y=-1/2x-5/2# #larr# perpendicular slope-intercept form

graph{(y-2x-3)(y+1/2x+5/2)=0 [-11.25, 11.25, -5.625, 5.625]}

Sadie Adams · Answer

undefined To find the equation of a line perpendicular to $ y = 2x + 3 $, we first determine the slope of the given line, which is $ m = 2 $.

The slope of a line perpendicular to $ y = 2x + 3 $ will have a slope that is the negative reciprocal of $ m $. So, the slope of the perpendicular line is $ -\frac{1}{2} $.

Using the point-slope form of the equation of a line, $ y - y_1 = m(x - x_1) $, where $ (x_1, y_1) $ is a point on the line and $ m $ is the slope, and substituting the given point $ (3, -4) $ and the slope $ -\frac{1}{2} $, we get:

$ y - (-4) = -\frac{1}{2}(x - 3) $

Simplifying:

$ y + 4 = -\frac{1}{2}x + \frac{3}{2} $

$ y = -\frac{1}{2}x + \frac{3}{2} - 4 $

$ y = -\frac{1}{2}x - \frac{5}{2} $

So, the equation of the line perpendicular to $ y = 2x + 3 $ and passing through $ (3, -4) $ is $ y = -\frac{1}{2}x - \frac{5}{2} $.