A national competition wants every round to have mean 50 and standard deviation 10, so that rounds…

Step 1. The multiplier. $\sigma_y = b\,\sigma_x$, so $10 = 8b$ and $b = 1.25$.

Step 2. The added amount. $50 = a + 1.25 \times 62 = a + 77.5$, so $a = -27.5$.

Step 3. A score of 70. $y = -27.5 + 1.25 \times 70 = 60$.

Step 4. The z-scores. Before: $\dfrac{70 - 62}{8} = 1$. After: $\dfrac{60 - 50}{10} = 1$. They are the same.

Why the same? A change with $b > 0$ moves and stretches every distance in the same way, so the number of SDs from the mean never changes.