Plug 'em in and see:

A. Plug in (x,y) = (-3,-3)

Plug it in the first inequality: 
      3x - y < 10
3(-3) - (-3) < 10
      -9 + 3 < 10
          -6 < 10

That's true.

Plug it in the second inequality:
      x + 6y > 15
(-3) + 6(-3) > 15
     -3 - 18 > 15
         -21 > 15

That's false.  So A isn't the answer because, even
though the first inequality is true, the seconds one
isn't, and in the correct choice, BOTH must be true.

--------------------

B. Plug in (x,y) = (9,1)

Plug it in the first inequality: 
      3x - y < 10
  3(9) - (1) < 10
      27 - 1 < 10
          26 < 10

That's false.  So there is no use to plug it in 
the second inequality for even if it would be 
true if we substituted it in the second inequality, 
it would not be the correct choice because it 
must give a true inequality when substituted 
in EITHER one!

-------------------------

C. Plug in (x,y) = (-6,6)

Plug it in the first inequality: 
      3x - y < 10
 3(-6) - (6) < 10
     -18 + 6 < 10
          -6 < 10

That's true.

Plug it in the second inequality:
      x + 6y > 15
 (-6) + 6(6) > 15
     -6 + 36 > 15
          30 > 15

That's true.  So C is a correct answer.

--------------------

The problem just states

"Which of the following ordered pairs satisfy both 3x - y < 10 
and x + 6y > 15?"

It doesn't say 

Which ONE of the following ordered pairs satisfy both 3x - y < 10 
and x + 6y > 15? 

So we have to see if D is also a correct answer as well:

------------------------

D. Plug in (x,y) = (7,-4)

Plug it in the first inequality: 
      3x - y < 10
 3(7) - (-4) < 10
      21 + 4 < 10
          25 < 10

That's false.   So now we know that there is only one
answer that is correct, and that is C.

Edwin