Your goal is to go from 3 equations in 3 variables down to 2 equations in 2 variables.
It is up to you to decide which variable you want to eliminate. I think I might try eliminating the y.
Add the first two equations together. since the y's are equal and opposite they will eliminate easily.
5x+4y-5z = 44
3x -4y +3z = -10
8x -2z = 34
since there is a common factor of 2, you can rewrite this equation as
4x-z =17 SAVE this EQ.
Since I haven't used the 3rd equation, I have to use it now. Since we have -3y in that equation, I will choose the first eq. to use again. That equation has a 4y. I am going to multiply by 3 and 4 to get 12y and -12y so they will cancel.
15x+12y -15z =132
24x -12y + 24z -60
39x -9z = 72
Recall the equation we saved:
4x - z = 17
I would multiply this equation by -9 and add it to the other one. That will eliminate z and you will solve for x.
Once you have x... find z in one of these two equations.
Once you have x, and z. Go back to one of the original equations and find y.
Once you have x, y, and z. Check all three values in all three equations. It is important to check because it is very, very easy to make a mistake. These are tricky and there are many places where you can make a mistake.