To solve the system of equations using the elimination method, we can eliminate the fractions by multiplying each equation by the least common multiple (LCM) of the denominators. In this case, the LCM of 7 and 6 is 42.
Multiply the first equation by 42 to get: 6x + 7y = 126
Multiply the second equation by 42 to get: 21x - 14y = 210
Now, we can eliminate the y term by multiplying the first equation by 2 and the second equation by 3.
2(6x + 7y) = 2(126) -> 12x + 14y = 252
3(21x - 14y) = 3(210) -> 63x - 42y = 630
Adding these two equations together, we get:
12x + 14y + 63x - 42y = 252 + 630
75x = 882
x = 882/75
x = 11.76
Substituting the value of x into the first equation, we can solve for y:
6(11.76) + 7y = 126
70.56 + 7y = 126
7y = 126 - 70.56
7y = 55.44
y = 55.44/7
y = 7.92
Therefore, the solution to the system of equations is x = 11.76 and y = 7.92.