Question

Solve the system of equations.

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

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,

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

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

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

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

Solving the equations we get,

$y+3z=1......\left(4\right)$

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

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

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

Solving the equations we get,

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

Solving equations (4) and (5).

Solving equations (4) and (5), we get,

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

Adding the equations we get,

$0=0$

The system of equations 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 $x+2y+z=4$

Substituting $y=1-3z$,

$$\begin{gathered} x+2(1-3z)+z=4 \\ x+2-6z+z=4 \\ x-5z=2 \\ x=2+5z. \end{gathered}$$