Final answer:
To find the sum of the x-coordinates of all possible positive integer solutions to the equation 1/x + 1/y = 1/7, we can rearrange the equation and use Simon's Favorite Factoring Trick to solve for x.
Step-by-step explanation:
To find the sum of the x-coordinates of all possible positive integer solutions to the equation 1/x + 1/y = 1/7, we can rearrange the equation to 7x + 7y = xy. Simplifying further, we have xy - 7x - 7y = 0. Applying Simon's Favorite Factoring Trick, we add 49 to both sides of the equation: xy - 7x - 7y + 49 = 49. Factoring the left side gives us (x - 7)(y - 7) = 49. Since we are looking for positive integer solutions, we can set x - 7 and y - 7 to be divisors of 49 and solve for x.
The divisors of 49 are 1, 7, and 49. Setting x - 7 = 1, we get x = 8. Setting x - 7 = 7, we get x = 14. And setting x - 7 = 49, we get x = 56. So the sum of the x-coordinates is 8 + 14 + 56 = 78.