Question

What is the equation of the line passing through the points `(3,3)` and `(-2, 17)`?

Answer 1

`y=-2.8x+11.4`

Explanation:

For any two points on a straight line (as given by a linear equation)
the ratio of the difference between the `y` coordinate values divided by the difference between the `x` coordinate values (called the slope) is always the same.

For the general point `(x,y)` and specific points `(3,3)` and `(-2,17)`
this means that:
the slope `=(Deltay)/(Deltax)=(y-3)/(x-3)=(y-17)/(x-(-2))=(3-17)/(3-(-2))`

Evaluating the last expression we have that
the slope `= (3-17)/(3-(-2))=(-14)/(5)=-2.8`

and therefore both
`{:((y-3)/(x-3)=-2.8,color(white)("XX")andcolor(white)("XX")(y-17)/(x-(-2))=-2.8):}`

We could use either of these to develop our equation; the first one seems easier to me (but feel free to test this with the second version to see that you get the same result).

If `(y-3)/(x-3)=-2.8`
then (assuming `x!=3`, otherwise the expression is meaningless)
after multiplying both sides by `(x-3)`
`color(white)("XX")y-3=-2.8x+8.4`
and therefore (after adding `3` to both sides)
`color(white)("XX")y=-2.8x+11.4`