Question

What is the equation of the line that passes through the point `(5, 5)` and is perpendicular to the line `5x+4y=36`?

Answer 1

`y=4/5x+1`

Explanation:

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

`color(red)(bar(ul(|color(white)(2/2)color(black)(y=mx+b)color(white)(2/2)|)))`
where m represents the slope and b, the y-intercept.

`"Rearrange " 5x+4y=36" into this form"`

`"Subtract 5x from both sides"`

`cancel(5x)cancel(-5x)+4y=-5x+36`

`rArr4y=-5x+36`

`"divide ALL terms by 4"`

`(cancel(4) y)/cancel(4)=-5/4 x+36/4`

`rArry=-5/4x+9larrcolor(red)" in slope-intercept form"`

`rArr"slope "=m=-5/4`

The slope of a perpendicular line to this line is the

`color(blue)"negative reciprocal" " of m"`

`rArrm_("perpendicular")=-1/(-5/4)=1xx4/5=4/5`

`"Using the " color(blue)"point-slope form"" of the equation"`

`• y-y_1=m(x-x_1)`

`"with "m=4/5" and " (x_1,y_1)=(5,5)`

`y-5=4/5(x-5)larrcolor(red)" in point-slope form"`

`"distribute and simplify"`

`y-5=4/5x-4`

`rArry=4/5x+1larrcolor(red)" in slope-intercept form"`