1. **State the problem:** Find the equation of the line passing through the points $(-3,9)$ and $(4,-15)$.\n\n2. **Formula used:** The slope $m$ of a line through points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $$m=\frac{y_2-y_1}{x_2-x_1}.$$\n\n3. **Calculate the slope:** Substitute the points into the formula: $$m=\frac{-15-9}{4-(-3)}=\frac{-15-9}{4+3}=\frac{-24}{7}.$$\n\n4. **Use point-slope form:** The equation of the line is $$y-y_1=m(x-x_1).$$ Using point $(-3,9)$, we get $$y-9=-\frac{24}{7}(x+3).$$\n\n5. **Simplify the equation:** Distribute the slope: $$y-9=-\frac{24}{7}x-\frac{24}{7}\times 3=-\frac{24}{7}x-\frac{72}{7}.$$\n\n6. **Isolate $y$:** $$y=-\frac{24}{7}x-\frac{72}{7}+9.$$\n\n7. **Convert 9 to sevenths:** $$9=\frac{63}{7},$$ so $$y=-\frac{24}{7}x-\frac{72}{7}+\frac{63}{7}=-\frac{24}{7}x-\frac{9}{7}.$$\n\n**Final answer:** $$y=-\frac{24}{7}x-\frac{9}{7}.$$