Question

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

Answer 1

`-2/5" and "5/2`

Explanation:

`"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`

Answer 2

`2/5 \ \ \` parallel.

`-5/2 \ \ \` perpendicular.

Explanation:

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)`