I can try - the best way to do these is to just practice - you'll start to get a more intuitive sense of what works and what doesn't. You start with:

2 -1 4 -3

1 0 -4 5

6 -1 2 10

If you can get two rows with 0s in the same column, it's often helpful. Since I have a 0 in R2, C2, I'm going to get a 0 in R3, C2. New R3 = R3 - R1. Note that the ONLY row I'm actually changing is R3, so R1 and R2 get copied exactly as they are.

2 -1 4 -3

1 0 -4 5

4 0 -2 13

Now, I can get another zero in R3 using a combination of R2. I'll get rid of the 4 in R3. New R3 = R3 - 4R2. Again, R1 and R2 don't change.

2 -1 4 -3

1 0 -4 5

0 0 14 -7

Let's finish R3 by dividing by 14.

2 -1 4 -3

1 0 -4 5

0 0 1 -.5

Now, using R3, I can easily eliminate anything I want in column 3 without messing anything else up. Start by cleaning up R2. New R2 = R2 + 4R3.

2 -1 4 -3

1 0 0 3

0 0 1 -.5

Now, let's clean up R1. New R1 = R1 - 4R3.

2 -1 0 -1

1 0 0 3

0 0 1 -.5

Clean up the first position in R1. New R1 = R1 - 2R2.

0 -1 0 -7

1 0 0 3

0 0 1 -.5

Rearrange and multiply the current R1 by -1.

1 0 0 3

0 1 0 7

0 0 1 -.5

So x = 3, y = 7, z = -.5

Let's check:

2x - y + 4z = -3

2(3) - 7 - 2 = 3. Check.

x - 4z = 5

3 + 2 = 5 Check.

6x - y + 2z = 10

6(3) - 7 - 1 = 10 Check.

Important things to remember:

Getting two zeros in the same column is incredibly useful because you can combine those rows to get a zero in an additional column without messing up what you've done. Once you've got a row with two zeros in it, you can pretty quickly clean up the rest. So, aim to get a zero in the same position in two different rows.

Also, as I learned while doing this, copy the problem down correctly.

Hope this helps!