Find the equation of the line joining (1, 2) and (2, 4). In this, find the sequence of y coordinates of those points with the consecutive natural numbers 3, 4, 5, … as the x coordinates.

x₁, x₂, x₃,… and y₁, y₂, y₃,… are arithmetic sequences. Prove that all points with coordinates in the sequence (x₁, y₁), (x₂, y₂), (x₃, y₃),… are on the same line.

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?

Find the equation of the line joining (–1, 3), (2, 5). Prove that if (x, y) is a point on this line, so is (x + 3, y + 2).

Prove that for any number x, the point (x, 2x + 3) is on the line joining (–1, 1), (2, 7).