Prove that if the point (x₁, y₁), (x₂, y₂), (x₃, y₃) are on a line, so are (3x₁ + 2y₁,3x₁ – 2y₁), (3x₂ + 2y₂,3x₂ – 2y₂), (3x₃ + 2y₃,3x₃ – 2y₃). Would this be true if we take other numbers instead of 3 and 2?

The three points are say A(x₁, y₁), B(x₂, y₂), C(x₃, y₃)

Since they lie on a line so slope of any two points are always equal

…Equation (i)

The other set of three points are say P(3x₁ + 2y₁,3x₁ – 2y₁), Q(3x₂ + 2y₂,3x₂ – 2y₂), R(3x₃ + 2y₃,3x₃ – 2y₃)

Since they also lie on a line so slope between any two points is always equal

Applying Componendo and dividendo we get

Applying Invertendo we get

…Equation (ii)

Since Equation (i) & Equation (ii) are similar so the points P,Q and R lie on the line joining A,B & C

Hence Proved

Yes it is possible if we take multiples of 2 and 3