I'm not quite sure where you're going with k vs k+1.

  we know that s + d = 209 (1)
  and we know that 6.5s + 2d = 836.5 (2)
  we can rewrite (1) as d = 209 - s (3)
  substituting (3) into (2) we  get 6.5s + 2(209 - s) = 836.5 (4)
  solving (4) for s yields s = 93 (5)

alternatively, you can solve the system of linear equations

  [   1 1 ] [ s ] = [   209 ]
  [ 6.5 2 ] [ d ] = [ 836.5 ] (1)
  
  [ s ] = [ -4/9  2/9 ] [   209 ]
  [ d ] = [ 13/9 -2/9 ] [ 836.5 ] (2)
  
  [ s ] = [ 93  ]
  [ d ] = [ 116 ] (3)

It's a really standard form of 2 formula substitution, you're missing the fact that there was 209 drinks and salads sold combined which constrains the solution to a single answer. From the setup you get 2 equations where x is the number of salads and y is the number of drinks:

> [1] 6.5x + 2 y = 836.5

and

> [2] x + y = 209

Solve [2] for y and substitute into [1]:

> 6.5*x + 2(209 - x) = 836.5

You can also do this with linear algebra but this is the way I would have known as a normal high schooler taking the SAT.