How do you find the slope of a line that is a) parallel and b) perpendicular to the given line: -2x-5y=-9?

#"the equation of a line in "color(blue)"slope-intercept form"# is.

#•color(white)(x)y=mx+b#

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

#"rearrange "-2x-5y=-9" into this form"#

#rArr-5y=2x-9#

#rArry=-2/5x+9/5larrcolor(blue)"in slope intercept form"#

#"with slope m "=-2/5#

#• " Parallel lines have equal slopes"#

#rArr"slope of parallel line is "m=-2/5#

#"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#

#rArrm_("perpendicular")=-1/(-2/5)=5/2#

First rearrange the equation to get the form:

#y=mx+b#

It is only in this form that we can see what the gradient is.

#-2x-5y=-9#

#y=2/5x+9/5#

If two lines have the same gradient, then they will be parallel. So any line of the form:

#y=2/5x+b# , will be parallel to # \ \ y=2/5x+9/5#

If two lines are perpendicular, then the product of their gradients is #-1#

For two lines with gradients #m_1# and #m_2#:

#m_1*m_2=-1#

#m_2=-1/m_1#

Let #m_1=2/5#

#:.#

#m_2=1/m_1=-1/(2/5)=-5/2#

So, any line of the form # bb(y=-5/2x+b)# will be perpendicular to #bb(y=2/5x+9/5)#