Question

-13 times the difference between a number and 60 is equal to the number plus -59

Answer 1

Let n represent the number.
  -13×(n -60) = n +(-59)
  -13n +780 = n -59
  839 = 14n
  839/14 = n = 59 13/14

The number is 59 13/14.

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Check
  The difference between the number and 60 is
    59 13/14 -60 = -1/14
  -13 times that value is
    -13*(-1/14) = 13/14
  Compare that to the number plus -59
    59 13/14 -59 = 13/14 . . . . . the same value, as required.