Question

How do you find the general form of the line perpendicular to 3x+5y-8=0 that passes through the point (-8,1)?

Answer 1

Therefore equation of the perpendicular line is:

`y=5/3x - 37/3`

Explanation:

Slope of perpendicular line is negative reciprocal of the original slope of the line.

So given linear equation is:
`3x+5y-8=0` -----> re-write this equation as `y=mx+b` where `m` is the slope of the line and `b` is the y-intercept.

`5y=8-3x`

`5y=-3x+8`

`y=-3/5y + 8/5` -----> Slope of the line is `-3/5` ----> `color(red)(RED -GRAPH)`

So the slope of the perpendicular line is `-1/(-3/5)` = `5/3`

So the equation of the perpendicular line is:
`y = 5/3x + b`

Lets us find `b` from the given points #(-8,1).

`y = 5/3x + b`

`1 = (5/3xx8) + b`

`1 = 40/3+b`

`b =1-40/3`

`b= -37/3`

Therefore equation of the perpendicular line is:

`y=5/3x - 37/3` ----> `color(blue) (BLUE -GRAPH)`

We can draw the graph and check as well (see the attached graphs)

Answer 2

`5x-3y+43=0`

Explanation:

`"the general form of the equation of a line is"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(Ax+By+C=0)color(white)(2/2)|)))`

`"where A, B and C are integers with A and B non-zero"`

`"obtain the equation in "color(blue)"slope-intercept form"`

`•color(white)(x)y=mx+b`

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

`"rearrange "3x+5y-8=0" into this form"`

`"subtract "3x-8" from both sides"`

`rArr5y=-3x+8`

`"divide all terms by 5"`

`rArry=-3/5x+8/5larrcolor(blue)"in slope-intercept form"`

`"with slope m "=-3/5`

`"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/(-3/5)=5/3`

`"now find the equation of the perpendicular line"`

`rArry=5/3x+blarrcolor(blue)"is the partial equation"`

`"to find b substitute "(-8,1)" into the partial equation"`

`1=-40/3+brArrb=3/3+40/3=43/3`

`rArry=5/3x+43/3larrcolor(red)"in slope-intercept form"`

`"rearrange into general form by multiplying all terms by 3"`

`rArr3y=5x+43`

`"subtract "3y" from both sides"`

`rArr5x-3y+43=0larrcolor(red)"in general form"`