Explanation

Divide both equations by xy and set u = 1/x, v = 1/y: 2u + 5v = 7 and 3u + 10v = 8. Doubling the first and subtracting the second gives u = 6, so x = 1/6; then 12 + 5v = 7 gives v = −1, so y = −1. Substituting x = 1/6 and y = −1 into both original equations verifies the solution.