Answer:
The solution is [tex]x=-1,\ y=-2,\ and\ z=0[/tex]
Step-by-step explanation:
Given:
The system of equations given are:
[tex]x-y+z=1\\x+y+3z=-3\\2x-y+2z=0[/tex]
Add first and second equations. This gives,
[tex]x-y+z+x+y+3z=1-3\\2x+4z=-2\\\textrm{Divide both sides by 2}\\x+2z=-1-----\ 4[/tex]
Adding equations 2 and 3, we get
[tex]x+y+3z+2x-y+2z=-3+0\\3x+5z=-3 --------------\ 5[/tex]
We need solve equations 4 and 5.
Multiplying equation (4) by -3, we get
[tex](x+2z=-1)\times -3=-3x-6z=3[/tex]
Adding the above equation to equation (5), we get
[tex]-3x-6z+3x+5z=3-3\\-z=0\\z=0[/tex]
Now, we plug in [tex]z=0[/tex] in equation (4). This gives,
[tex]x+2(0)=-1\\x+0=-1\\x=-1[/tex]
Now, we plug in [tex]x=-1,z=0[/tex] in equation (1). This gives,
[tex]-1-y+0=1\\-1-y=1\\-y=1+1\\-y=2\\y=-2[/tex]
Therefore, the solution is [tex]x=-1,\ y=-2,\ and\ z=0[/tex]
