1. **State the problem:** We have two functions $f(x) = 5x + 7$ and $g(x) = 6 - 8x$. We want to find $h(x)$ where $h(x) = f(x) + g(x)$ and then evaluate $h(6)$. Also, find the value of $k$ such that $h(5) = f(4) + g(k)$. 2. **Find $h(x)$:** $$h(x) = f(x) + g(x) = (5x + 7) + (6 - 8x)$$ Simplify: $$h(x) = 5x + 7 + 6 - 8x = (5x - 8x) + (7 + 6) = -3x + 13$$ 3. **Evaluate $h(6)$:** $$h(6) = -3(6) + 13 = -18 + 13 = -5$$ 4. **Find $k$ such that $h(5) = f(4) + g(k)$:** Calculate $h(5)$: $$h(5) = -3(5) + 13 = -15 + 13 = -2$$ Calculate $f(4)$: $$f(4) = 5(4) + 7 = 20 + 7 = 27$$ Set up equation: $$h(5) = f(4) + g(k) \\ -2 = 27 + g(k) \\ g(k) = -2 - 27 = -29$$ Recall $g(k) = 6 - 8k$, so: $$6 - 8k = -29$$ Solve for $k$: $$-8k = -29 - 6 = -35$$ $$k = \frac{\cancel{-35}}{\cancel{-8}} = \frac{35}{8}$$ **Final answers:** - $h(6) = -5$ - $k = \frac{35}{8}$