I’m having serious problems in comprehending matrix row reduction. I know how to turn an equation into a matrix and just to clarify I think these are the rules:

1. I can switch any two rows at any given point.

2. I can multiply or divide a row by any number of my choosing.

3. I can add or subtract any two rows with each other. The resulting answer is then replaced into the row that should change (ex: row1+row2 means row 2 is replaced. If row1-row2 then row1 is replaced).

4. I should be concentrating on an outcome of:

1 0 0 x

0 1 0 y

0 0 1 z

5. If I'm able to get x, y or z separately by itself I can use regular algebra to solve for the rest. (Ex: I have 1 0 0 x but the rows of y and z still don't look as they should. I can simply plug in what x is with the linear equations given above to solve for y and z.)

By this point, I don't understand the logical steps that should be taken when solving a problem. In other words, when I see examples I do not getwhysomeone is choosing to multiply a row or subtract a row. Could someone step me through how I would solve:

2x-y+4z=-3

x-4z=5

6x-y+2z=10

Thanks.