Question

Solve the system of equations.

$\left\{\begin{array}{l}x+y-2z=3\\ -2x-3y+z=-7\\ x+2y+z=4\end{array}\right.$

Short answer

The solution for the system of equations is$(5z+2,1-3z,z)$.

Step-by-step solution

Step 1. Given the information.

The system of equations is,$\left\{\begin{array}{l}x+y-2z=3..........\left(1\right)\\ -2x-3y+z=-7....\left(2\right)\\ x+2y+z=4..........\left(3\right)\end{array}\right.$

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

Eliminating $x$from the equations (1) and (2).

$$\begin{gathered} 2x+2y-4z=6.......\left(1\right)\times 2 \\ -2x-3y+z=-7....\left(2\right) \end{gathered}$$

Solving the equations we get,

$$\begin{gathered} -y-3z=-1 \\ y+3z=1.......\left(4\right) \end{gathered}$$

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

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

$$\begin{gathered} -2x-3y+z=-7.......\left(2\right) \\ 2x+4y+2z=8..........\left(3\right)\times 2 \end{gathered}$$

Solving the equations we get,

$y+3z=1.........\left(5\right)$

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

Solving equations (4) and (5),

$$\begin{gathered} y+3z=1.........\left(4\right) \\ y+3z=1.........\left(5\right) \end{gathered}$$

Subtracting the equations we get,

$0=0$

The solution has infinitely many solutions.

Step 5. Showing how y is dependent on z.

Simplifying the equation $y+3z=1$,

$$\begin{gathered} y+3z=1 \\ y=1-3z \end{gathered}$$

Step 6. Showing how x is dependent on z.

Simplifying the equation $x+y-2z=3$

Substituting $y=1-3z$

$$\begin{gathered} x+y-2z=3 \\ x+(1-3z)-2z=3 \\ x+1-3z-2z=3 \\ x+1-5z=3 \\ x-5z=2 \\ x=2+5z \end{gathered}$$