The sum of two numbers is 7 five times the larger number plus four times the the smaller number is 47.

Let x and y represent the larger and smaller numbers, respectively. The problem conditions give rise to two equations:
  x + y = 7
  5x +4y = 47

You can use the first equation to write an expression for y, then substitute that into the second equation.
  y = 7 - x
  5x +4(7 -x) = 47 . . . . substitute for y
  x + 28 = 47 . . . . . . .. simplify
  x = 19 . . . . . . . . . . . . subtract 28

Using our expression for y, we have
  y = 7 -19 = -12

The larger number is 19.
The smaller number is -12.