(1) 7x - 4y = 5
(2) 21x - 12y = 15

We need to manipulate one or both equations in such a way that, when added, one of the variable terms sums to zero.

Look at eqn 2. We could get the x-variable terms to sum to zero we could get the 21x to become -7x. To do this, let's divide each term of eqn 2 by negative 3 (same as mult by negative one third) to give a new eqn 2

(-1/3)[21x - 12y = 15] = -7x + 4y = -5

(1)          7x - 4y = 5
(new 2) -7x + 4y = -5


Summing the two eqns gives

0x + 0y = 0

0 = 0

In this case, we have an identity. This means the two equations describe the same line (they would be on top of one another when graphed).  As opposed to the two lines intersecting at a unique point (one soln) they share every point (infinite solns). 

If you end up with a statement like

3 = 1

The statement is false, and there are no solutions (lines never intersect, meaning parallel slopes). Such a systems is said to be "inconsistent."