Question

What is the equation of the line with slope `m= 13/7` that passes through `(7/5,4/7)`?

Answer 1

`65x-35y=71`

Explanation:

Given a slope `m` and a point `(barx,bary)`
the "slope-point form" of the linear equation is
`color(white)("XXX")(y-bary)=m(x-barx)`

Given
`color(white)("XXX")m=13/7`
and
`color(white)("XXX")(barx,bary)=(7/5,4/7)`

The "slope-point form" would be:
`color(white)("XXX")(y-4/7)=13/7(x-7/5)`
and this should be a valid answer to the given question.

However, this is ugly, so let's convert it into standard form:
`color(white)("XXX")Ax+By=C` with `A, B, C in ZZ, A>=0`

Multiply both sides by `7`
`color(white)("XXX")7y-4=13x-91/5`

Multiply both sides by `5` to clear the remaining fraction
`color(white)("XXX")35y-20=65x-91`

Subtract `(35y-91)` from both sides to get the variables on one side and the constant on the other
`color(white)("XXX")71=65x-35y`

Exchange sides:
#color(white)("XXX")65x-35y=71