Question

Line L has equation 2x - 3y = 5.

Line M passes through the point (2, -10) and is perpendicular to line L.
Determine the equation for line M.

Answer 1

y=-
`(3)/(2)`x-13

Explanation:

You have to convert everything to slope intercept form before starting

2x-3y=5

3y=2x-5

y=2/3x-5/3

Then,

Since Line M is perpendicular Line M's slope will be,

-
`(3)/(2)`x

y=-3/2x+b

Enter the given point (2,-10)

-10=-3/2(2)+b

-10=3+b

b=-10-3

b=-13

so the equation for line M is

y=-3/2x-13

Answer 2

`y=-(3)/(2)x -7`

Explanation:

Line L has the equation:

`2x-3y=5`

we need to clear for y:

`-3y=-2x+5\\y=(2)/(3) x-(5)/(3)`

now we have the form a general line equation

`y=mx+b`

where
`m` is the slope of the line.

so the slope of the line L is:

`m_(1)=(2)/(3)`

and for two lines to be parallel the following condition must be met

`m_(1)*m_(2)=-1`

where
`m_(2)` in this case is the slope of line M, substituting the value
`m_(1)` to find
`m_(2)`:

`(2)/(3)*m_(2)=-1\\ m_(2)=(-1(3))/(2)\\m_(2)=-(3)/(2)`

This is the slope of line M, and since we also know that it passes through the point (2, -10) where
`x_(0)=2` and
`y_(0)=-10`

we use the point- slope equation and substitute known values to find the equation of the line M:

`y-y_(0)=m(x-x_(0))\\y-(-10)=-(3)/(2)(x-2)\\ y+10=-(3)/(2)x+3\\y=-(3)/(2)x +3-10\\y=-(3)/(2)x -7`

the equation of line M is:

`y=-(3)/(2)x -7`