Question

What is the equation of the line with slope `m= 17/3` that passes through `(7/9,8/3)`?

Answer 1

In slope point form: `(y-8/3) = (17/3)(x-7/9)`

In standard form: `153x-27y = 47`

Explanation:

The general slope-point form for a line with slope `m` through a point `(hatx,haty)` is
`color(white)("XXX")(y-haty) = m(x-hatx)`

For the given values this becomes:
`color(white)("XXX")(y-8/3)=(17/3)(x-7/9)`

To convert this to standard form we will need to do some simplification.

Begin clearing the denominators by multiplying both sides by `3`
`color(white)("XXX")3y-8 = 17(x-7/9)`
Continue clearing the denominators by multiplying both sides by `9`
`color(white)("XXX")27y-72 = 17(9x-7) = 153x-119`

Subtract `(153x)` from both sides
`color(white)("XXX")-153x + 27y -72 = -119`

Add `72` to both sides
`color(white)("XXX")-153x+27y = -47`

Multiply both sides by `(-1)`
`color(white)("XXX")153x-27y = 47`