1. **State the problem:** Find the correct equation(s) of the line passing through the point $(-1,4)$ with slope $m = -3$. 2. **Formula used:** The point-slope form of a line is $$y - y_1 = m(x - x_1)$$ where $(x_1, y_1)$ is a point on the line and $m$ is the slope. 3. **Apply the formula:** Using point $(-1,4)$ and slope $-3$, $$y - 4 = -3(x - (-1)) = -3(x + 1)$$ This matches option B. 4. **Check option A:** $$y = -\frac{1}{4}x - 3$$ Slope here is $-\frac{1}{4}$, which does not match $-3$, so A is incorrect. 5. **Check option C:** $$y = -3x + 1$$ Check if it passes through $(-1,4)$: $$4 \stackrel{?}{=} -3(-1) + 1 = 3 + 1 = 4$$ True, so C is correct. 6. **Check option D:** $$3x + y = 1$$ Rewrite in slope-intercept form: $$y = -3x + 1$$ This is the same as option C, so D is also correct. **Final answer:** Options B, C, and D are correct equations for the line.