1. The problem is to find the equation of the line passing through the points $(4,-3)$ and $(3,-6)$.\n\n2. The formula to find the slope $m$ of a line through two points $(x_1,y_1)$ and $(x_2,y_2)$ is:\n$$m=\frac{y_2-y_1}{x_2-x_1}$$\n\n3. Substitute the given points into the slope formula:\n$$m=\frac{-6-(-3)}{3-4}=\frac{-6+3}{3-4}=\frac{-3}{-1}=3$$\n\n4. Now use the point-slope form of the line equation:\n$$y-y_1=m(x-x_1)$$\nChoose point $(4,-3)$:\n$$y+3=3(x-4)$$\n\n5. Simplify the equation:\n$$y+3=3x-12$$\n$$y=3x-12-3$$\n$$y=3x-15$$\n\n6. The equation of the line passing through $(4,-3)$ and $(3,-6)$ is:\n$$y=3x-15$$