1. **Problem Statement:** Given the ratio $$(x - 2y + 8z) : (y - 3z + 4w) : (4x + 5z - 7w) = 3 : 4 : 7,$$ find the value of $$\frac{6x + 2y - 17w}{w}.$$ 2. **Understanding the problem:** The expressions are in ratio form, meaning there exists a constant $k$ such that: $$x - 2y + 8z = 3k,$$ $$y - 3z + 4w = 4k,$$ $$4x + 5z - 7w = 7k.$$ 3. **Goal:** Express $6x + 2y - 17w$ in terms of $w$ and $k$ to find the ratio asked. 4. **Step 1: Express variables in terms of $k$ and $w$:** We have three equations: $$x - 2y + 8z = 3k \quad (1)$$ $$y - 3z + 4w = 4k \quad (2)$$ $$4x + 5z - 7w = 7k \quad (3)$$ 5. **Step 2: Solve for $x, y, z$ in terms of $k$ and $w$:** From (2): $$y = 4k + 3z - 4w.$$ Substitute $y$ into (1): $$x - 2(4k + 3z - 4w) + 8z = 3k,$$ $$x - 8k - 6z + 8w + 8z = 3k,$$ $$x + 2z + 8w = 11k,$$ so, $$x = 11k - 2z - 8w.$$ 6. **Step 3: Substitute $x$ and $y$ into (3):** $$4(11k - 2z - 8w) + 5z - 7w = 7k,$$ $$44k - 8z - 32w + 5z - 7w = 7k,$$ $$44k - 3z - 39w = 7k,$$ $$-3z = 7k - 44k + 39w = -37k + 39w,$$ $$z = \frac{37k - 39w}{3}.$$ 7. **Step 4: Substitute $z$ back to find $x$ and $y$:** $$x = 11k - 2\times \frac{37k - 39w}{3} - 8w = 11k - \frac{74k - 78w}{3} - 8w = \frac{33k - 74k + 78w - 24w}{3} = \frac{-41k + 54w}{3}.$$ $$y = 4k + 3\times \frac{37k - 39w}{3} - 4w = 4k + 37k - 39w - 4w = 41k - 43w.$$ 8. **Step 5: Calculate $6x + 2y - 17w$:** $$6x + 2y - 17w = 6 \times \frac{-41k + 54w}{3} + 2(41k - 43w) - 17w = 2(-41k + 54w) + 82k - 86w - 17w,$$ $$= -82k + 108w + 82k - 103w = 5w.$$ 9. **Step 6: Divide by $w$:** $$\frac{6x + 2y - 17w}{w} = \frac{5w}{w} = 5.$$ **Final answer:** 5