1. The problem asks to identify the correct graph of the linear equation $$y + 3 = -\frac{1}{2}(x - 4)$$. 2. First, rewrite the equation in slope-intercept form $y = mx + b$ to understand the slope and y-intercept. 3. Start by expanding the right side: $$y + 3 = -\frac{1}{2}x + 2$$ 4. Subtract 3 from both sides to isolate $y$: $$y = -\frac{1}{2}x + 2 - 3$$ 5. Simplify the constant terms: $$y = -\frac{1}{2}x - 1$$ 6. The slope $m$ is $-\frac{1}{2}$, which means the line slopes downward gently. 7. The y-intercept $b$ is $-1$, so the line crosses the y-axis at $(0, -1)$. 8. Check the point $(4, -3)$ given in the graphs: Substitute $x=4$: $$y = -\frac{1}{2}(4) - 1 = -2 - 1 = -3$$ This matches the point $(4, -3)$. 9. Check the slope between $(4, -3)$ and another point on the line to confirm: Using $(6, -2)$: Slope $= \frac{-2 - (-3)}{6 - 4} = \frac{1}{2} = 0.5$ 10. The slope from the equation is $-\frac{1}{2} = -0.5$, but the slope between $(4, -3)$ and $(6, -2)$ is positive $0.5$, so these points do not lie on the line. 11. Check the slope between $(4, -3)$ and $(1, -4)$: Slope $= \frac{-4 - (-3)}{1 - 4} = \frac{-1}{-3} = \frac{1}{3} \approx 0.33$ This is positive, not matching the negative slope. 12. Check the slope between $(4, -3)$ and $(0, 0)$: Slope $= \frac{0 - (-3)}{0 - 4} = \frac{3}{-4} = -0.75$ This is negative and steeper than $-0.5$. 13. The slope from the equation is $-0.5$, so the closest slope is $-0.5$. 14. Therefore, the correct graph is the one with slope $-\frac{1}{2}$ passing through $(4, -3)$ and matching the equation. 15. Graph B has a negative slope steeper than $-\frac{1}{2}$, Graph C has a more negative slope, Graph A and D have positive slopes. 16. Hence, none of the graphs perfectly match the slope $-\frac{1}{2}$ and points given except the point $(4, -3)$. 17. Since Graph B has a negative slope and passes through $(4, -3)$ and $(1, -4)$, but slope is $\frac{1}{3}$, not $-\frac{1}{2}$, it is incorrect. 18. Graph C has slope $-0.75$, which is closer but steeper than $-0.5$. 19. Graph A and D have positive slopes, so they are incorrect. 20. The correct graph is the one with slope $-\frac{1}{2}$ and passing through $(4, -3)$, which is the original equation. 21. To find a second point on the line, use the slope $-\frac{1}{2}$ from $(4, -3)$: From $x=4$ to $x=6$ (increase by 2), $y$ decreases by $2 \times \frac{1}{2} = 1$: $$y = -3 - 1 = -4$$ 22. So the point $(6, -4)$ lies on the line. 23. None of the graphs show $(6, -4)$, so the correct graph is not among the options exactly. 24. Summary: The line has slope $-\frac{1}{2}$ and passes through $(4, -3)$ and $(6, -4)$. Final answer: The correct graph corresponds to the line $$y = -\frac{1}{2}x - 1$$ passing through $(4, -3)$ and $(6, -4)$, which is not exactly any of the given graphs.