This is a beautiful problem based on Solving Equations from Test of Mathematics Subjective Problem no. 20.
Problem : Solving equations
If \(\ a,b,c,d\) satisfy the equations
$$a+7b+3c+5d=0,$$
$$8a+7b+6c+2d=-16,$$
$$2a+6b+4c+8d=16,$$
$$5a+3b+7c+d=-16,$$
then \(\ (a+d)(b+c)\) equals
\(\ (A)16 \quad (B)-16\quad (C)0 \quad\) (D)none of the foregoing numbers
Solution:
$$a+7b+3c+5d=0\dots(1),$$
$$8a+7b+6c+2d=-16\dots(2),$$
$$2a+6b+4c+8d=16\dots(3),$$
$$5a+3b+7c+d=-16\dots(4),$$
\(\ (1)-(3)\), and \(\ (2)-(4)\), we get
$$-a+b-c-3d=-16\dots(5),$$
$$3a+b-c+d=0\dots(6),$$
\(\ (6)-(5)\), we get
$$a+d=4\dots(7),$$
\(\ (2)+(3)\),we get
$$a+b+c+d=0\dots(8),$$
\(\ (8)-(7)\),we get
$$b+c=-4\dots(9),$$
\(\ (7)\times(9)\),we get
Therefore,$$(a+d)(b+c)=-16$$
Thus,\(\ (B)\) is the correct option.
