To solve the equation \( x - 7 = 1 \) for \( x \), we want to isolate \( x \) on one side of the equation.
To do this, we can add 7 to both sides of the equation.
So:
\[ x - 7 + 7 = 1 + 7 \]
Simplifying both sides gives:
\[ x = 8 \]
Therefore, the number that should be added to both sides of the equation is \( 7 \).