Answer:
x = 6
Explanation:
The angle bisector divides the triangle sides proportionally:
(6x -1)/42 = (10x +5)/78
13(6x -1) = 7(10x +5) . . . . multiply by 546 (the LCM of 42 and 78)
78x -13 = 70x +35 . . . . . eliminate parentheses
8x = 48 . . . . . . . . . . . . . . add 13-70x
x = 6 . . . . . . . . . . . divide by 8
_____
Additional comment
The unknown segment lengths are ...
6·6 -1 = 35
10·6 +5 = 65
Then the ratios to the known sides are ...
35/42 = 65/78 = 5/6