-2x -3y + z = -6

X+y-z =5 

7x + 8y -6z=31

We have the following system of three linear equations:

\[
\begin{aligned}
1) & \quad -2x - 3y + z = -6 \\
2) & \quad x + y - z = 5 \\
3) & \quad 7x + 8y - 6z = 31
\end{aligned}
\]

### Step 1: Solve for one variable
Let's solve Equation 2 for \(z\):

\[
x + y - z = 5 \quad \Rightarrow \quad z = x + y - 5
\]

Now, substitute \(z = x + y - 5\) into Equations 1 and 3.

### Step 2: Substitute into Equations 1 and 3

Substitute \(z = x + y - 5\) into Equation 1:
\[
-2x - 3y + (x + y - 5) = -6
\]
Simplify:
\[
-2x - 3y + x + y - 5 = -6
\]
\[
-x - 2y = -1 \quad \Rightarrow \quad x + 2y = 1 \quad \text{(Equation 4)}
\]

Substitute \(z = x + y - 5\) into Equation 3:
\[
7x + 8y - 6(x + y - 5) = 31
\]
Simplify:
\[
7x + 8y - 6x - 6y + 30 = 31
\]
\[
x + 2y = 1 \quad \text{(Equation 5, same as Equation 4)}
\]

### Step 3: Solve for \(x\) and \(y\)

Since Equation 4 and Equation 5 are identical, they represent a single equation:
\[
x + 2y = 1
\]

Let's express \(x\) in terms of \(y\):
\[
x = 1 - 2y
\]

### Step 4: Substitute back to find \(z\)

Substitute \(x = 1 - 2y\) into the expression for \(z\) from Equation 2:
\[
z = (1 - 2y) + y - 5
\]
Simplify:
\[
z = 1 - 2y + y - 5 = -4 - y
\]

### Final Solution

\[
x = 1 - 2y, \quad y = y, \quad z = -4 - y
\]

This is the parametric form of the solution, where \(y\) is a free variable. The general solution is:
\[
\boxed{(x, y, z) = (1 - 2y, y, -4 - y)}
\]
where \(y\) can be any real number.

### Verification
- For \(y = 0\), \(x = 1\), and \(z = -4\).
- For \(y = 1\), \(x = -1\), and \(z = -5\).
- Substituting these into the original equations will confirm their validity.

Would you like more details or have any questions?

### Related Questions
1. How do you solve systems of linear equations with dependent variables?
2. What is the significance of having a parametric solution in linear systems?
3. How does Gaussian elimination work for solving systems of equations?
4. What are some examples of systems of equations with no solutions?
5. Can you explain how to interpret the geometric meaning of a system of linear equations?

### Tip
In systems with multiple variables, consider the possibility of free variables, which might indicate infinitely many solutions.

Learn how to solve the system of linear equations -2x -3y + z = -6, x + y - z = 5, 7x + 8y - 6z = 31 step-by-step. This problem involves finding parametric solutions and covers concepts in linear algebra. Suitable for high school students.

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