Question

Solve the system:$\left\{\begin{array}{c}x-2y+z=15\\ 3x+y-3z=-8\\ -2x+4y-z=-27\end{array}\right.$

Short answer

The solution of the given system:

$$\begin{gathered} x=2 \\ y=-5 \\ z=3 \end{gathered}$$

Step-by-step solution

Step 1. Given information.

System of equations:

$\left\{\begin{array}{c}x-2y+z=15.........equation\left(1\right)\\ 3x+y-3z=-8...........equation\left(2\right)\\ -2x+4y-z=-27...........equation\left(3\right)\end{array}\right.$

Step 2. Add equation (1) and (3)

$$\begin{gathered} x-2y+z-2x+4y-z=15-27 \\ -x+2y=-12 \end{gathered}$$

$x-2y=12$..........equation(4)

Step 3. Substitute the value of equation (4) in to the equation (1).

$$\begin{gathered} 12+z=15 \\ z=15-12 \\ z=3 \end{gathered}$$

Step 4. Substitute value of z into the equation (2).

$$\begin{gathered} 3x+y-3\times 3=-8 \\ 3x+y-9=-8 \\ 3x+y=9-8 \\ 3x+y=1 \end{gathered}$$

$y=1-3x$...........equation (5)

Step 5. Substitute the value of y from equation (5) into the equation (4).

$$\begin{gathered} x-2(1-3x)=12 \\ x-2+6x=12 \\ 7x-2=12 \\ 7x=12+2 \\ 7x=14 \\ x=\frac{14}{7} \\ x=2 \end{gathered}$$

Step 6. Substitute the value of x into the equation (5).

$$\begin{gathered} y=1-3\times 2 \\ y=1-6 \\ y=-5 \end{gathered}$$