14. (15 points) Four people, A, B, C, and D, go fishing and catch a total of 25 fish. Ranked by the number of fish caught from most to least, it is A, B, C, and D. It is also known that the number of fish A caught is the sum of the number of fish B and C caught, and the number of fish B caught is the sum of the number of fish C and D caught. How many fish did A, B, C, and D each catch?
【Solution】Solution: Let's assume that A, B, C, and D caught $a$, $b$, $c$, and $d$ fish respectively, and $a>b>c>d$. According to the problem, we can get the equation: $a+b+c+d=25$, (1);
$a=b+c$, (2);
$b=c+d$, (3);
Substituting (3) into (2) gives: $a=2c+d$, (4);
Substituting (3) and (4) into (1) gives:
$4c+3d=25$, solving this system of linear equations for integer solutions yields:
When $c=1$, $d=7$;
When $c=4$, $d=3$;
Also, because: $c>d$, the only solution that fits the problem is $c=4, d=3$,
So $b=4+3=7; a=7+4=11$;
Answer: A, B, C, and D caught 11, 7, 4, and 3 fish respectively.