The THIRD sentence is correct.
The first equation can be multiplied by -12 and the second equation by 5, to eliminate x.
Let's do this:
(1st equation) 5x + 13y = 232 / (-12)
(2nd equation) 12x + 7y = 218 / (5)
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When multiplied, we get:
(1st equation) -60x - 156y = -2784
(2nd equation) 60x + 35y = 1090
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Further, first equation now will be obtained when ADDING 1st and 2nd equation together:
(1st equation) -60x+60x-156y+35y=-2784+1090
2nd equation we get from just writing any equation from the beginning, for example, the second one:
(2nd equation) 12x + 7y = 218
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(1st equation) 0x - 121y = -1694
(2nd equation) 12x + 7y = 218
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(1st equation) -121y = -1694 [x variable is in this step eliminated]
(2nd equation) 12x + 7y = 218
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(1st equation) y = 1694/121
(2nd equation) 12x + 7y = 218
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(1st equation) y = 14
(2nd equation) 12x + 714 = 218
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(1st equation) y = 14
(2nd equation) 12x + 98 = 218
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(1st equation) y = 14
(2nd equation) 12x = 218 - 98
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(1st equation) y = 14
(2nd equation) 12x = 120
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(1st equation) y = 14
(2nd equation) x = 120/12
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(1st equation) y = 14
(2nd equation) x = 10
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So, solution of the system of this two equations is obtained, and it is:
(x,y)=(10,14)
