Question

What is the equation of the line perpendicular to `y=-7/16x` that passes through `(5,4)`?

Answer 1

`y=16/7x-52/7`

See details below

Explanation:

If a line has the equation `y=mx`, we call slope to `m` and whatever perpendicular line to it has then the equation `y=-1/mx`

In our case `y=-7/16x`, then, the slope is `m=-7/16`, so the perpendicular has slope `m´=-1/(-7/16)=16/7`. Our perpendicular line is

`y=16/7x+b`. But this line passes through `(5,4)`. Then

`4=16/7·5+b`. Transposing terms we have `b=-52/7`

Finally, perpendicular line equation is `y=16/7x-52/7`

Answer 2

`y=16/7x-52/7`

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"`

`y=-7/16x" is in this form"`

`"with "m=-7/16`

`"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/(-7/16)=16/7`

`rArry=16/7x+blarrcolor(blue)"is the partial equation"`

`"to find b substitute "(5,4)" into the partial equation"`

`4=80/7+brArrb=28/7-80/7=-52/7`

`rArry=16/7x-52/7larrcolor(red)"perpendicular equation"`