If x + 5 y = 4 and y + 5 x = -4 , then what is (x + y) equal to?

Step-by-step solution

Given equations are x + 5 y = 4 and y + 5 x = -4 . Adding the two equations, we get: x + y + 5 x + 5 y = 0 (x + y) + 5 ( x + y xy ) = 0 (x + y) (1 + 5 xy ) = 0 This implies either x + y = 0 or 1 + 5 xy = 0 . Subtracting the second equation from the first, we get: x - y - 5 x + 5 y = 8 (x - y) + 5 ( x - y xy ) = 8 (x - y) (1 + 5 xy ) = 8 If 1 + 5 xy = 0 , the second factor becomes zero, which gives 0 = 8 , a contradiction. Thus, 1 + 5 xy 0 . Therefore, we must have x + y = 0 . Answer: 0