Answer
The new score:
$X=13$
Work Step by Step
$M=\frac{∑X}{N}$
Before:
There are 7 scores whose mean is $5$:
$5=\frac{X_1+X_2+X_3+X_4+X_5+X_6+X_7}{7}$
$X_1+X_2+X_3+X_4+X_5+X_6+X_7=5\times7=35$
After:
There are 8 scores whose mean is $6$:
$6=\frac{X_1+X_2+X_3+X_4+X_5+X_6+X_7+X_8}{8}$
$X_1+X_2+X_3+X_4+X_5+X_6+X_7+X_8=6\times8=48$
$35+X_8=48$
$X=48-35=13$
