If w, x, y, and z are real numbers satisfying
w + x + y = -2
w + x + z = 4
w + y + z = 29 and
x + y + z = 22
what is xz + yz?
+129847 w + x + y = -2 ⇒ x + y = -2 - w (1)
w + x + z = 4 ⇒ x + z = 4 - w (2)
w + y + z = 29 ⇒ y + z = 29 - w (3)
x + y + z = 22 ⇒ 2x + 2y + 2z = 44 (4)
Add (1) , (2) , (3)
2x + 2y + 2z = 31 - 3w (5)
Sub (5) into (4)
31 - 3w = 44
-3w = 44 - 31
-3w = 13
w = -13/3
And using (1)
x + y = -2 - (-13/3) = 7/3
x + y + z = 22
(7/3) + z = 22
z = 22 - 7/3 = 59/3
So
xz + yz =
(x + y)z =
(7/3) (59/3) =
413 / 9
