Question

Solve the system of equations,

$\left\{\begin{array}{l}x-2y+2z=1\\ -2x+y-z=2\\ x-y+z=5\end{array}\right.$

Short answer

The system of equations has no solution.

Step-by-step solution

Step 1. Given the information.

The system of equations is,

$\left\{\begin{array}{l}x-2y+2z=1......\left(1\right)\\ -2x+y-z=2.....\left(2\right)\\ x-y+z=5.........\left(3\right)\end{array}\right.$

Step 2. Eliminating x from equations (1) and (2).

Eliminating $x$from equations (1) and (2),

$$\begin{gathered} 2x-4y+4z=2..........\left(1\right)\times 2 \\ -2x+y-z=2...........\left(2\right) \end{gathered}$$

Solving the equations we get,

role="math" localid="1644567307021" $-3y+3z=4........\left(4\right)$

Step 3. Eliminating x from equations (2) and (3).

Eliminating $x$from equations (2) and (3).

$$\begin{gathered} -2x+y-z=2........\left(2\right) \\ 2x-2y+2z=10.....\left(3\right)\times 2 \end{gathered}$$

Solving the equations we get,

$-y+z=12.......\left(5\right)$

Step 4. Solving equations (4) and (5).

Solving equations (4) and (5),

$$\begin{gathered} -3y+3z=4.......\left(4\right) \\ -3y+3z=36......\left(5\right)\times 3 \end{gathered}$$

Subtracting the equations we get,

$0=-32$

The system of equations has no solution.