Fran works due north of home. Her husband Alan works due east. By the time Fran is 7 miles from home, the distance between them is one mile more than Alan's distance from home. How far from home is Alan? Miles

Did you mean "walks" instead of "works" ?
Not that it matters much, I get a right-angled triangle with sides 7, x, and x+1
(x+1)^2 = x^2 + 7^2
x^2 + 2x + 1 = x^2 + 49
2x = 48
x = 24