Of . 3x+4y+4z=4, 5x+7y+3z=5 4x+5y+7z=7

We'll label the equations to keep track of what we're doing

3x+4y+4z=4    A

5x+7y+3z=5    B

4x+5y+7z=7     C

Normally in Gaussian elimination we'd pick put a constant on each equation and combine to cancel out a variable in all but one.  But here we have an opportunity to make the coefficient 1 on our variables in at least one equation.  This is a pretty good thing to do when solving these by hand.

x + y + 3z = 3         D=C-A

-2x - 3y + z = -1     E=A-B

Let's eliminate x from two equations

-y + 7z = 5          F=2D+E

-y + 5z = 5         G=4D-C

Let's eliminate y

2z = 0               H=F-G

z = 0

-y + 7z = 5

y = -5

x + y + 3z  = 3

x - 5 + 0  = 3

x = 8

(x,y,z)=(8,-5,0)

Check:

3(8)+4(-5)+4(0) = 4, good

5(8)+7(-5)+3(0)=5, good

4(8)+5(-5)+7(0)=7, good