Find the equation of the line passing through the point $(-2, 3)$ with a slope of $-4$.

The straight line $L$ is perpendicular to the line $5x - y = 1$. If the area of the triangle formed by the line $L$ and the coordinate axes is $5$,find the equation of the line $L$.

Find the slope of the line making an inclination of $60^{\circ}$ with the positive direction of the $x$-axis.

Find the equation of a straight line passing through $(-5, 6)$ and cutting off equal intercepts on the coordinate axes.

Find the value of $k$ for which the line $(k-3) x - (4-k^2) y + k^2 - 7k + 6 = 0$ passes through the origin.

Find the equation of the line such that the perpendicular drawn from the origin to the line makes an angle of $30^{\circ}$ with the $x$-axis and the line forms a triangle of area $\frac{50}{\sqrt{3}}$ with the axes.