One leg of a right triangle is 7 foot more than the other leg. the hypotenuse is 13 feet. find the lengths of the two legs.

Let n = our shorter leg
n + 7 = longer leg
13 = hypotenuse

The Pythagorean Theorem is a² + b² = c² where a and b are legs and c is the hypotenuse. Now, we plug in our given information, solve for n, then plug in the result of n into our leg measurements. 

n² + (n + 7)² = 13²
n² + n² + 14n + 49 = 169
2n² + 14n + 49 - 169 = 0
2n² + 14n - 120 = 0

Remove a GCF of 2 from each term in the trinomial
n² + 7n - 60 = 0
(n - 5)(n + 12) = 0

Set each binomial equal to zero
n - 5 = 0
n + 12 = 0

n = - 12, 5

A measurement cannot be negative so n = 5 is the one we use. 

Your short leg, n, is 5 feet. 
Your longer leg, n + 7, is 12 feet
And your hypotenuse is 13 given by the description.