+147 If $\frac{1}{x} + \frac{1}{y} = 3$ and $\frac{1}{x} - \frac{1}{y} = -7$ what is the value of $x + y$? Express your answer as a common fraction.
+128631 1/x + 1/y = 3 → ( y + x) / xy = 3 → (y + x) / 3 = xy (1)
1/x - 1/y = -7 → (y - x) / xy = - 7 → ( y - x) / -7 = xy (2)
Setting (1) , (2) equal....we have that
(y + x) / 3 = (y - x) / -7 cross-multiply
-7 ( x +y) = 3 ( y - x)
-7x - 7y = 3y - 3x
-10y = 4x
y = (-4/10) x
y = ( -2/5 ) x....sub this back into 1/x + 1/y = 3
1/x + 1/ [(-2/5) x ] = 3
1/x - 5/2x = 3
( 1 -5/2) /x = 3
(-3/2) / 3 = x
-1/2 = x
So
y = (-2/5) (-1/2) = 1/5
x + y = -1/2 + 1/5 = - 5/10 + 2/10 = -3 / 10
