−4x + y = 6
−5x − y = 21

You have in the first equation "+y" and in the second "-y"
So, you add the 2 equations, so we can remove this 2 letters since they're opposite numbers.
So, we get:
-4x + y -5x - y = 21 + 6
-4x -5x = 21 + 6

Then, you add
-4x -5x = 21 + 6
-9x = 27

Then, you move the -9 to the other part, next to the 27. And since it was a multiplication, it becomes a division.
x = 


Then, you choose an equation of the system, and you replace x by it's value so you can find y. I'll take the first equation.

-4x + y = 6
-4*-3 + y = 6
12 + y = 6

Then, you move 12 to the other part next to the 6. And since it was a positive number, it becomes a negative number.
So
y = 6 - 12
y = -6

So, the solution of the proposed system is (x;y) = (6;-6)
x = 6
y = -6

Hope this Helps :)