Let's have A as the hypotenuse, B as the long leg and C as the short leg, as you put it.
A = 2C + 3 from The length of the hypotenuse is 3cm more than double that of the shorter leg.
B = C + 7 from the length of one leg of a right triangle is 7cm more than that of the other leg.
Since Pythagoras' theorem applies we know that A^2 = B^2 + C^2.
ie. (2C + 3)^2 = (C+7)^2 + C^2
4C^2 + 12C + 9 = C^2 + 14C + 49 + C^2
2C^2 - 2C -40 = 0
C^2 - C -20 = 0.
(C - 5)(C + 4) = 0.
So C = 5 or -4.
And since you can't have a negative length, C = 5.
And if C = 5, B=C+7=12, A = 2C+3 = 13.