You mean like graph? Yeah.
Write in Function form
[tex]\frac{y-3}{x-6}=2\\y-3=2(x-6)\\y=2(x-6)+3\\y=2x-12+3\\y=2x-9[/tex]
So (y-3/x-6)=2 does represent "line"
[tex]\frac{y-6}{x-4}=5[/tex]
multiply x-4 all side.
[tex]y-6=5(x-4)[/tex]
Notice how the equation is similar to "Tangent" so it's clearly a line.
About reasoning, No idea how to explain but this might help...
Whenever you get rid of the fraction part, it'll leave only y-f(a)=f'(a)(x-a) which is a Tangent Line (It explains it.)
And when you simplify in Function Form, you'll get an exact linear function "[tex]y=mx+b[/tex]"
