linear algebra - Proof of elementary row operations for matrices?

I'm taking a Linear Algebra course, and we just started talking about matrices. So we were introduced to the elementary row operations for matrices which say that we can do the following:

1. Interchange two rows.


2. Multiply a row with a nonzero number.


3. Add a row to another one multiplied by a number.

Now I understood from the lecture in class how to use these and all, but I want to understand the logic behind number 3.
Is there a mathematical proof that shows that by adding row $R_1$ to row $R_2$ we are not changing the system of equations?

Thanks in advance