Question

How do you write the equation that represents the line perpendicular to `y=-3x +4` and passing through the point (-1,1)?

Answer 1

`y = 1/3x + 4/3`

Explanation:

Step 1) Because this is already solved for `y` we can get the slope of the perpendicular line by taking the slope of the given line (-3) and "flipping" it and reversing the sign:

`-3` becomes `- -1/3 -> + 1/3 -> 1/3`

Therefore the slope of the perpendicular line is `1/3`

Step 2) Use the point slope for to find the perpendicular equation:

`y - 1 = 1/3(x - -1)`

`y - 1 = 1/3(x + 1)`

`y - 1 = 1/3x + 1/3`

`y - 1 + 1 = 1/3x + 1/3 + 1`

`y - 0 = 1/3x + 1/3 + 3/3`

`y = 1/3x + 4/3`

Answer 2

`y=(1/3)x+(4/3)`

Explanation:

Any linear equation of the form
`color(white)("XXX")y=color(green)mx+color(blue)b`
has a slope of `color(green)m`

Any line perpendicular to a line with slope `color(green)m`
has a slope `color(green)(""(-1/m))`

`y=color(green)(-3)x+4` has a slope of `color(green)(-3)`
`rarr` any line perpendicular to it has a slope `color(green)(1/3)`

A line with slope `color(magenta)k` through the point `(color(red)a,color(blue)b)`
can be written in slope-point form as:
`color(white)("XXX")(y-color(blue)b)=color(magenta)k(x-color(red)a)`

Therefore the line perpendicular to `y=-3x+4` and through the point `(color(red)(-1),color(blue)1)`
can be written as:
`color(white)("XXX")y-color(red)1=color(green)(1/3)(x-color(red)(""(-1)))`

While this is a perfectly valid answer to the given question, we would normally convert this into standard form as:
`color(white)("XXX")x-3y=-4`
or slope-vertex form (as was the initial equation):
`color(white)("XXX")y=1/3x+4/3`