Question

How do you write an equation in slope intercept form perpendicular to y=1/2x-4, contains (4,1)?

Answer 1

`y=-2x+9`

Explanation:

An equation in the form `y=color(green)mx+color(blue)b` is in slope intercept form with a slope of `color(green)m`

Therefore `y=color(green)(1/2)x-4` has a slope of `color(green)(1/2)`

If a line has a slope of `color(green)m` then any line perpendicular to it has a slope of `color(magenta)(-1/m)`

Therefore any line perpendicular to `y=color(green)(1/2)x-4`
must have a slope of `color(magenta)(-2)`
and a slope-intercept form of `y=(color(magenta)(-2))x+color(blue)b` for some constant `color(blue)b`

If `(color(red)x,color(orange)y)=(color(red)4,color(orange)1)` is to be a point on the required perpendicular line then
`color(white)("XXX")color(orange)1=(color(magenta)(-2)) * color(red)4 +color(blue)b`

`color(white)("XXX")rarr color(blue)b=9`

and the equation of the required perpendicular line is
`color(white)("XXX")y=color(magenta)(-2)x+color(blue)9`

Answer 2

`y=-2x+9`

Explanation:

`y=1/2x-4" is in "color(blue)"slope-intercept form"`

`"That is " y=color(red)(m)xcolor(blue)(+b) " where " color(red)(m)` represents the slope and `color(blue)(+b)` the y-intercept.

`rArr" slope "=m=1/2`

The slope of a line perpendicular to this line is `color(blue)" the negative reciprocal"` of `color(red)(m)`

`rArrm_("perpendicular")=-1/(1/2)=-2`

The equation can be written partially as `y=-2x+b`

To calculate b substitute (4 ,1) into the partial equation.

`rArr(-2xx4)+b=1`

`rArrb=1+8=9`

`rArry=-2x+9larrcolor(red)" in slope-intercept form"`