Elimination method.
Transform one or both equation so that a variable cancels out when combined together. Same as substitution method, we're trying to come up with one equation with only one variable.
Multiplying the second equation by -3,
-3(x - 5y = -22) --> -3x + 15y = 66
Combine this with the second equation.
3x + 4y = 22
-3x + 15y = 66
----------------------- add
19y = 88 iff
y = 88/19
Like before, plug in y = 88/19 into any of the two equations and solve for x.
x - 5y = -22
x - 5(88/19) = -22 iff
x = 22/19
Thus the solution is (22/19, 88/19), same as above.