Question

Solve:x + 2y + z = 82x + y - z = 1x + y – 2z = - 3

Answer 1

We are given the following system of equations:

`\begin{gathered} x+2y+z=8,(1) \\ 2x+y-z=1,(2) \\ x+y-2z=-3,(3) \end{gathered}`

To solve the system we will add equations (1) and (2):

`x+2y+z+2x+y-z=8+1`

Adding like terms:

`\begin{gathered} 3x+3y=9 \\ x+y=3,(4) \end{gathered}`

Now we multiply equation (2) by -2:

`-4x-2y+2z=-2`

Now we add this equation to equation (3):

`x+y-2z-4x-2y+2z=-3-2`

Adding like terms:

`-3x-y=-5,(5)`

Now we add equations (4) and (5):

`x+y-3x-y=3-5`

Adding like terms:

`-2x=-2`

Dividing both sides by -2:

`x=-(2)/(-2)=1`

Now we replace this value of "x" in equation (4):

`\begin{gathered} x+y=3 \\ 1+y=3 \end{gathered}`

Subtracting 1 to both sides:

`\begin{gathered} 1-1+y=3-1 \\ y=2 \end{gathered}`

Now we replace the values of "x" and "y" in equation (1):

`\begin{gathered} x+2y+z=8 \\ 1+2(2)+z=8 \end{gathered}`

Adding like terms:

`\begin{gathered} 1+4+z=8 \\ 5+z=8 \end{gathered}`

Subtracting 5 to both sides:

`\begin{gathered} 5-5+z=8-5 \\ z=3 \end{gathered}`

Therefore, the solution of the system is:

`x=1,\text{ y=2, z=3}`