1. **State the problem:** We want to find the value of $a$ for which the system of equations $$-x + 6y = 7$$ $$-5x + 10ay = 32$$ has no solution. 2. **Recall the condition for no solution:** A system of two linear equations has no solution if the lines are parallel but not coincident. This means their ratios of coefficients of $x$ and $y$ are equal, but the ratio of constants is different. 3. **Write the ratios:** For the system $$\frac{-1}{-5} = \frac{6}{10a} \neq \frac{7}{32}$$ 4. **Simplify the ratios:** $$\frac{-1}{-5} = \frac{1}{5}$$ and $$\frac{6}{10a} = \frac{3}{5a}$$ Set the first two ratios equal for parallel lines: $$\frac{1}{5} = \frac{3}{5a}$$ 5. **Solve for $a$:** Multiply both sides by $5a$: $$5a \times \frac{1}{5} = 5a \times \frac{3}{5a}$$ $$a = 3$$ 6. **Check the constants ratio:** $$\frac{7}{32} \neq \frac{1}{5}$$ So the lines are parallel but not coincident, meaning no solution. **Final answer:** $a = 3$