Question

find the complete solution of the linear system, or show that the system is inconsistent. (if the system has no solution, enter no solution.)
\\(\

$$\begin{cases}x - 2y+3z=-14\\3y + z = 5\\x + y - z = 11\\end{cases}$$

\\)
\\((x,y,z)=(4,3, - 4)\\)

Explanation:

Step1: Label the equations

Let \(x - 2y+3z=-14\) be Equation (1), \(3y + z = 5\) be Equation (2), and \(x + y - z=11\) be Equation (3).

Step2: Eliminate \(x\)

Subtract Equation (1) from Equation (3): \((x + y - z)-(x - 2y+3z)=11-(-14)\).
Simplify to get \(3y-4z = 25\), call it Equation (4).

Step3: Solve for \(z\) using Equation (2) and (4)

We have the system \(

$$\begin{cases}3y + z = 5\\3y-4z = 25\end{cases}$$

\). Subtract Equation (4) from Equation (2): \((3y + z)-(3y-4z)=5 - 25\).
\(5z=-20\), so \(z=-4\).

Step4: Solve for \(y\)

Substitute \(z = - 4\) into Equation (2): \(3y-4 = 5\), then \(3y=9\), and \(y = 3\).

Step5: Solve for \(x\)

Substitute \(y = 3\) and \(z=-4\) into Equation (3): \(x+3-(-4)=11\), \(x + 7=11\), so \(x = 4\).

Answer:

\((x,y,z)=(4,3,-4)\)