1. **Stating the problem:** 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 ratio:** The given ratio means there exists a constant $k$ such that: $$ x - 2y + 8z = 3k $$ $$ y - 3z + 4w = 4k $$ $$ 4x + 5z - 7w = 7k $$ 3. **Express variables in terms of $k$ and $w$:** We want to find $\frac{6x + 2y - 17w}{w}$, so it is convenient to express $x$ and $y$ in terms of $w$ and $k$. 4. **From the second equation:** $$ y - 3z + 4w = 4k \implies y = 4k + 3z - 4w $$ 5. **From the first equation:** $$ x - 2y + 8z = 3k $$ Substitute $y$: $$ x - 2(4k + 3z - 4w) + 8z = 3k $$ $$ x - 8k - 6z + 8w + 8z = 3k $$ $$ x + 2z + 8w = 3k + 8k $$ $$ x + 2z + 8w = 11k $$ $$ x = 11k - 2z - 8w $$ 6. **From the third equation:** $$ 4x + 5z - 7w = 7k $$ Substitute $x$: $$ 4(11k - 2z - 8w) + 5z - 7w = 7k $$ $$ 44k - 8z - 32w + 5z - 7w = 7k $$ $$ 44k - 3z - 39w = 7k $$ $$ -3z = 7k - 44k + 39w $$ $$ -3z = -37k + 39w $$ $$ z = \frac{37k - 39w}{3} $$ 7. **Substitute $z$ back into $x$ and $y$:** $$ x = 11k - 2\left(\frac{37k - 39w}{3}\right) - 8w = 11k - \frac{74k - 78w}{3} - 8w $$ $$ x = \frac{33k}{3} - \frac{74k - 78w}{3} - \frac{24w}{3} = \frac{33k - 74k + 78w - 24w}{3} = \frac{-41k + 54w}{3} $$ $$ y = 4k + 3\left(\frac{37k - 39w}{3}\right) - 4w = 4k + 37k - 39w - 4w = 41k - 43w $$ 8. **Calculate numerator $6x + 2y - 17w$:** $$ 6x + 2y - 17w = 6\left(\frac{-41k + 54w}{3}\right) + 2(41k - 43w) - 17w $$ $$ = 2(-41k + 54w) + 82k - 86w - 17w $$ $$ = -82k + 108w + 82k - 103w = 5w $$ 9. **Divide by $w$:** $$ \frac{6x + 2y - 17w}{w} = \frac{5w}{w} = 5 $$ **Final answer:** $$ \boxed{5} $$