Forming the statement mathematically(And ignoring the constant C since we're taking only the x and y coordinates):
Let $y = mx$ be our line
Let $(x_1, y_1)$ and $(x_2, y_2)$ be the two points that satisfy the equation.
Hence, we can write the statements as:
$y_1 = m\cdot x_1$ and $y_2 = m\cdot x_2$
Now, how do we prove that:
The equation is satisfied for the following:
$(ax_1+bx_2, ky_1+ly_2)$ i.e.
$(ky_1+ly_2) = m(ax_1+bx_2)$
where a, b, k and l are constants.