To rewrite the equation \( a = 10b - 7 \) in terms of \( b \), Arthur needs to isolate \( b \) on one side of the equation.
First, he should move \( -7 \) to the other side. To do this, he should add \( 7 \) to both sides of the equation. This gives:
\[ a + 7 = 10b \]
So, the value that Arthur should add to both sides of the equation is 7.