In this system of equations, you have 3 equations and 3 unknowns. The top equation has +y, and the middle has -y. That means you can add the middle equation to the top equation to eliminate y. You get [x+y+2z]+[2x-y+3z]=[-2]+[-11]. Combine like terms to get 3x+5z=-13
Now add the middle equation to the bottom equation: [2x-y+3z]+[3x+y-2z]=[-11]+[4]. Combine like terms to get 5x+z=-7. You can solve this for z by subtracting 5x from both sides to get z=-7-5x. Now plug this into the equation you found earlier to get 3x+5[-7-5x]=-13. Distribute the 5 to get 3x-35-25x=-13. Combine like terms to get -22x-35=-13. Add 35 to both sides to get -22x=22. Now divide both sides by -22 to get x=-1. At this point, you can plug that value into your earlier equation to get 3(-1)+5z=-13. Distribute to get -3+5z=-13. Add 3 to both sides to get 5z=-10. Now divide by 5 to get z=-2. At this point, you can plug in the values of x and z into your original equation to get [-1]+y+2[-2]=-2. Distribute the 2 to get -1+y-4=-2. Combine like terms to get y-5=-2. Finally, add 5 to both sides to get y=3. You now have your answer: x=-1, y=3, z=-2.
