Here's how you solve it: (wall of text)
x+2y+z=4
4y-3z=1
2y+10z=12
Step 1. Isolate variables
(math stuff here from second equation)
x=4-z-2y
y=(3z+1)/4
z=-1/3+4/3y
Step 2. Substitute into other equation to solve a variable
Let's solve z, shall we?
2y+10z=12
substitute in y
2((3z+1)/4)+10z=12
1.5z+10z+.5=12
isolate Z
11.5z=11.5
divide by 11.5 to get
z=1
Step 3. Substitute 1 for z to solve other variables
y=(3z+1)/4
y=4/4
y=1
x=4-z-2y
x=4-1-2
x=1
Solution is (1,1,1)!