How the heck do you solve these type of problems?

Do the math the same way you would if all the other variables were numbers. For linear equations such as these, you can
• subtract one side of the equation, so you get an equation equal to zero.
• factor out the variable of interest
• subtract all terms that don't have the variable as a factor
• divide by the coefficient of the variable.

19. 2xy - x - 7 = 0 . . . . subract the right side
  x(2y-1) -7 = 0 . . . . factor out x
  x(2y -1) = 7 . . . . subtract the term(s) not containing x. Subtract -7 here.
  x = 7/(2y -1) . . . . divide by the coefficient of x

20. Usually for these, you would eliminate fractions first. (You don't have to, but it does simplify things a bit.)
  2d + f = 24 . . . . multiply the equation by 4 to eliminate fractions
  f = 24 - 2d . . . . subtract the term not involving f

21. We can go back to the above sequence of steps here, if you like.
  3(2j -k) -108 = 0 . . . . subtract the right side
  6j -3k -108 = 0 . . . . separate the j terms from the others (eliminate parentheses using the distributive property)
  6j = 108 + 3k . . . . add the opposite of the non-j terms
  j = 108/6 + 3k/6 . . . . divide by the coefficient of j
  j = 18 + k/2 . . . . simplify

22. "solve for" variable not specified.
  t = (3.5s -42)/4
  s = (4t+42)/3.5
In both cases, the 3.5 can be replaced by 7/2 and the fraction simplified.
  t = (7/8)(s-12)
  s = 12 + (8/7)t