Question

How do you write an equation in slope-intercept form for a line with (4, -1) and perpendicular to a line with equation 3x + y = 4?

Answer 1

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

Explanation:

A linear equation in the form `Ax+By=C` has a slope of `-A/B` [Note 1, below]

The given line `3x+y=4` has a slope of `-3/1=-3`

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If two lines are perpendicular the slope of one is the negative reciprocal of the other.

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

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A line with a slope of `color(green)m` through a point `(color(red)(x_1),color(blue)(y_1))`
can be written in slope-point form as:
`color(white)("XXX")y-color(blue)(y_1)=color(green)m(x-color(red)(x_1))` [Note 2, below]

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

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Here is a graph with these two equations and the required point for verification:

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Note 1
If you are not familiar with slope of `Ax+By=C` being `-A/B`
it can be easily derived by converting to slope intercept form:
`color(white)("XXX")Ax+By=C`
`color(white)("XXX")rarr By=-Ax+C`
`color(white)("XXX")rarr y=-A/Bx+C/B` with slope of `-A/B` and y-intercept `C/B`

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Note 2
For some the more common general slope-point form might be
`color(white)("XXX")((y-color(blue)y_1))/(x-color(red)((x_1)))=color(green)m`
I prefer the form that I have used above since it avoids the exclusion problem when `x=color(red)(x_1)`