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
Answer: (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
