In what ratio does the $x$-axis divide the line segment joining the points $(-4, -6)$ and $(-1, 7)$? Find the coordinates of the point of division.

(6:7) Let the required ratio be $\lambda : 1$. The coordinates of the point $M$ dividing the line segment joining $A(-4, -6)$ and $B(-1, 7)$ are given by the section formula:
$\left( \frac{\lambda x_2 + 1 \cdot x_1}{\lambda + 1}, \frac{\lambda y_2 + 1 \cdot y_1}{\lambda + 1} \right)$
Substituting the values $x_1 = -4, x_2 = -1, y_1 = -6, y_2 = 7$:
$\left( \frac{\lambda(-1) + 1(-4)}{\lambda + 1}, \frac{\lambda(7) + 1(-6)}{\lambda + 1} \right) = \left( \frac{-\lambda - 4}{\lambda + 1}, \frac{7\lambda - 6}{\lambda + 1} \right)$
Since the line segment is divided by the $x$-axis,the $y$-coordinate of the point of division must be $0$:
$\frac{7\lambda - 6}{\lambda + 1} = 0 \implies 7\lambda - 6 = 0 \implies \lambda = \frac{6}{7}$
Thus,the required ratio is $6:7$.
Now,substituting $\lambda = \frac{6}{7}$ into the $x$-coordinate expression:
$x = \frac{-\frac{6}{7} - 4}{\frac{6}{7} + 1} = \frac{-\frac{34}{7}}{\frac{13}{7}} = -\frac{34}{13}$
Therefore,the point of division is $\left( -\frac{34}{13}, 0 \right)$.