The mid-point of the line segment joining the points (3m, 6) and (-4, 3n) is (1, 2m - 1). Find the values of m and n.

We know by mid-point formula,

$x = \dfrac{x$1 + x2}{2} \text{ and } y = \dfrac{y1 + y2}{2}

Given, the mid-point of (3m, 6) and (-4, 3n) is (1, 2m - 1).

$\therefore \text{x-coordinate } = 1 = \dfrac{3m + (-4)}{2} \\[1em] \Rightarrow 1 = \dfrac{3m - 4}{2} \\[1em] \Rightarrow 3m - 4 = 2 \\[1em] \Rightarrow 3m = 2 + 4 \\[1em] \Rightarrow 3m = 6 \\[1em] \Rightarrow m = 2.$

Similarly by midpoint formula,

$\text{y-coordinate } = 2m - 1 = \dfrac{6 + 3n}{2}$

Putting value of m = 2 in above equation we get,

$\Rightarrow 2 \times 2 - 1 = \dfrac{6 + 3n}{2} \\[1em] \Rightarrow 2(4 - 1) = 6 + 3n \\[1em] \Rightarrow 6 = 6 + 3n \\[1em] \Rightarrow 3n = 0 \\[1em] \Rightarrow n = 0.$

Hence, the value of m = 2 and n = 0.