This is not a regular type of problem where you have n equations and n unknowns and where there is a unique solution. You could write equations like x = 7 * k + 3 where x is the total number of fruit and k is what each of seven people gets, but then you get three equations with four unknowns. However, there is an additional restriction that k is an integer, which is not taken into account when systems of equations are solved in a regular way. Even with this restriction, there are infinitely many solutions.
The correct way to solve this type of problems is to use the Chinese remainder theorem. (See Dr. Math and Wikipedia.) Unfortunately, to understand the proof you probably need to know modular arithmetic.
The problem asks for an x such that
x leaves a remainder 3 when divided by 7
x leaves a remainder 7 when divided by 11
x leaves a remainder 1 when divided by 13
Chinese remainder theorem is applicable here because 7, 11 and 13 are pairwise coprime. Then one can show that there exist numbers r1, r2, r3 and p1, p2, p3 such that
r1 * 7 + p1 * 11 * 13 = 1
r2 * 11 + p2 * 7 * 13 = 1
r3 * 13 + p3 * 7 * 11 = 1
One set of solutions is
r1 = 41; p1 = -2
r2 = -33; p2 = 4
r3 = 6; p3 = -1
Let e1 = p1 * 11 * 13 = -286, e2 = p2 * 7 * 13 = 364, e3 = p3 * 7 * 11 = -77. Then
e1 leaves a remainder 1 when divided by 7 and a remainder 0 when divided by 11 and 13
e2 leaves a remainder 1 when divided by 11 and a remainder 0 when divided by 7 and 13
e3 leaves a remainder 1 when divided by 13 and a remainder 1 when divided by 7 and 11
So, one can take x = 3 * e1 + 7 * e2 + 1 * e3 = 1613. One can also add and subtract 7 * 11 * 13 = 1001 at will. In particular, 1613 - 1001 = 612 is another solution.