1. **State the problem:** Find the missing number $a$ in the equation $a x - 2 - 5x = -2x + 10$ so that the equation has no solutions. 2. **Rewrite the equation:** Combine like terms on the left side: $$a x - 5x - 2 = -2x + 10$$ This simplifies to: $$ (a - 5)x - 2 = -2x + 10$$ 3. **Bring all terms to one side:** $$ (a - 5)x - 2 + 2x - 10 = 0$$ Simplify: $$ (a - 5 + 2)x - 12 = 0$$ $$ (a - 3)x - 12 = 0$$ 4. **Analyze for no solutions:** For a linear equation $mx + b = 0$ to have no solutions, the equation must be a contradiction, meaning the variable terms cancel out but the constants do not. So, set the coefficient of $x$ to zero: $$a - 3 = 0 \implies a = 3$$ Then the equation becomes: $$0 \cdot x - 12 = 0 \implies -12 = 0$$ which is false, so no solutions exist. 5. **Final answer:** The missing number is $\boxed{3}$.