What is the equation of the line perpendicular to #y = 3x -2# and passes through point (3,4)?

2 Answers
Oct 9, 2017

#y=-1/3x+5#

Explanation:

#"given a line with slope m then the slope of a line "#
#"perpendicular to it is"#

#•color(white)(x)m_(color(red)"perpendicular")=-1/m#

#y=3x-2" is in "color(blue)"slope-intercept form"#

#•color(white)(x)y=mx+b#

#"where m is the slope and b the y-intercept"#

#y=3x-2rArrm=3#

#rArrm_(color(red)"perpendicular")=-1/3#

#rArry=-1/3x+blarr" is the partial equation"#

#"to find b substitute "(3,4)" into the partial equation"#

#4=-1+brArrb=5#

#rArry=-1/3x+5larrcolor(red)" in slope-intercept form"#

Oct 9, 2017

#y=color(red)(-1/3)x+color(blue)(5)#

Explanation:

Let us take the equation of the line as:
#y=color(red)(m)x+color(blue)(b)# , where m is the slope of the line.

Since the 2 lines are perpendicular, we use the negative reciprocal rule where:

  • The product of the lines' slopes should be equal to -1.

From this rule, we get that:
#m*3=-1#

We divide both sides by 3 to get the slope #m#:
#mcancel(*3)/cancel 3=-1/3#
#-> color(red)(m=-1/33#

Equation of line:
#y=-1/3x+b#

To find the y-intercept #b#, we will have to put the values:
Let us call the given point A:
#A(3,4)#

Since A lies on the line:
#y_A=-1/3*x_A+b#
#4=-1/cancel3cancel(*3)+b#
#4=-1+b#

Adding 1 to both sides to isolate y-intercept b:
#4+1=cancel(-1)cancel(+1)+b#
#-> color(blue)(b=5#

Putting the values of #color(red)(m# and #color(blue)(b#
#y=color(red)(m)x+color(blue)(b)#
#rArry=color(red)(-1/3)x+color(blue)(5)#