To solve this system of equations, we can use substitution or elimination method. Let's use substitution method:

From the first two equations:
AB + BC = 9cm
AC + BC = 13cm

Solve the first equation for AB:

AB = 9 - BC

Now substitute this expression for AB into the third equation:

(9 - BC) + AC + BC = 14cm
9 + AC = 14cm
AC = 5cm

Now substitute AC = 5cm into the second equation to solve for BC:

5 + BC = 13cm
BC = 8cm

Finally, substitute AC = 5cm and BC = 8cm into the first equation to solve for AB:

AB + 8 = 9cm
AB = 1cm

Therefore, AB = 1cm, AC = 5cm, and BC = 8cm.