Question

How do you write an equation of a line in function notation given the line goes through (2,3); perpendicular to `6x-7y=6`?

Answer 1

Background for slopes:
`color(white)("XXXX")`The slope of a line is defined as
`color(white)("XXXX")` `color(white)("XXXX")` `(Delta y)/(Delta x)`
`color(white)("XXXX")`That is, given two points `(x_1,y_1)` and `(x_2,y_2)` on the line
`color(white)("XXXX")`the slope is
`color(white)("XXXX")` `color(white)("XXXX")` `m=(y_2-y_1)/(x_2-x_1)`

`color(white)("XXXX")`For a straight line the slope is the same for all pairs of points on the line
`color(white)("XXXX")`Therefore, given two fixed points (as above) and a variable point `(x,y)` on the line
`color(white)("XXXX")` `color(white)("XXXX")` `(y-y_1)/(x-x_1) = (y_2-y_1)/(x_2-x_1)`
`color(white)("XXXX")`This can be rewritten:
`color(white)("XXXX")` `color(white)("XXXX")` `y=m(x-x_1)+y_1`

`color(white)("XXXX")`If a line has a slope of `hatm` then all lines perpendicular to it have a slope of `1/(hatm)`

Slope of `6x-7y=6`
`color(white)("XXXX")`This equation can be rewritten as
`color(white)("XXXX")` `color(white)("XXXX")` `y = (6/7)x+(6/7)`
`color(white)("XXXX")`and therefore has a slope of `(6/7)`
`color(white)("XXXX")`Any line perpendicular to it has a slope of `(-7/6)`

Equation of a line through `(2,3)` perpendicular to `6x-7y=6`
`color(white)("XXXX")`Using the previous discussion:
`color(white)("XXXX")` `color(white)("XXXX")` `y = (-7/6)(x-2)+3`
`color(white)("XXXX")`or, simplified and re-written in function notation
`color(white)("XXXX")` `color(white)("XXXX")` `f(x) = -7/6x+2/3`