Solve the following system of equations for all three variables.

x, plus, y, minus, z, equals, minus, 2

*x*+*y*−*z*=

−2

minus, 3, x, plus, 9, y, minus, 5, z, equals, 6

−3*x*+9*y*−5*z*=

6

x, plus, 9, y, minus, 7, z, equals, minus, 8

*x*+9*y*−7*z*=

−8

To solve the given system of equations for the variables \(x\), \(y\), and \(z\), we can use either substitution, elimination, or matrix methods (like Gaussian elimination). The system of equations is:

\[
\begin{cases}
x + y - z = -2 \quad \text{(Equation 1)} \\
-3x + 9y - 5z = 6 \quad \text{(Equation 2)} \\
x + 9y - 7z = -8 \quad \text{(Equation 3)}
\end{cases}
\]

### Step 1: Simplify Equations

**Equation 1:**
\[
x + y - z = -2
\]

**Equation 2:**
\[
-3x + 9y - 5z = 6
\]

**Equation 3:**
\[
x + 9y - 7z = -8
\]

### Step 2: Use Elimination Method

We can start by manipulating the equations to eliminate one variable at a time.

#### Multiply Equation 1 by 3:

\[
3(x + y - z) = 3(-2)
\]

Simplifies to:

\[
3x + 3y - 3z = -6 \quad \text{(Equation 4)}
\]

#### Add Equation 2 and Equation 4:

\[
(-3x + 9y - 5z) + (3x + 3y - 3z) = 6 + (-6)
\]

Simplifies to:

\[
12y - 8z = 0
\]

Simplify further by dividing by 4:

\[
3y - 2z = 0 \quad \text{(Equation 5)}
\]

#### Subtract Equation 1 from Equation 3:

\[
(x + 9y - 7z) - (x + y - z) = -8 - (-2)
\]

Simplifies to:

\[
8y - 6z = -6
\]

Simplify by dividing by 2:

\[
4y - 3z = -3 \quad \text{(Equation 6)}
\]

### Step 3: Solve Equations 5 and 6 Simultaneously

Now, solve Equations 5 and 6:

\[
\begin{cases}
3y - 2z = 0 \\
4y - 3z = -3
\end{cases}
\]

**Multiply Equation 5 by 3:**

\[
9y - 6z = 0 \quad \text{(Equation 7)}
\]

**Multiply Equation 6 by 2:**

\[
8y - 6z = -6 \quad \text{(Equation 8)}
\]

Subtract Equation 7 from Equation 8:

\[
(8y - 6z) - (9y - 6z) = -6 - 0
\]

Simplifies to:

\[
-y = -6
\]

\[
y = 6
\]

### Step 4: Substitute \( y = 6 \) into Equation 5

\[
3(6) - 2z = 0
\]

\[
18 - 2z = 0
\]

\[
2z = 18
\]

\[
z = 9
\]

### Step 5: Substitute \( y = 6 \) and \( z = 9 \) into Equation 1

\[
x + 6 - 9 = -2
\]

\[
x - 3 = -2
\]

\[
x = 1
\]

### Final Solution

\[
x = 1, \quad y = 6, \quad z = 9
\]

Would you like more details on any step, or do you have any questions?

Here are some related questions you might find interesting:

1. How can Gaussian elimination be used to solve this system?
2. What happens if the equations are dependent or inconsistent?
3. How can matrix methods like Cramer's rule be applied to this problem?
4. How would the solution change if the constants in the equations were altered?
5. What is the geometric interpretation of solving this system?

**Tip:** Always check your solutions by substituting them back into the original equations to verify their correctness!

Solve the following system of equations for all three variables:

x + y - z = -2
-3x + 9y - 5z = 6
x + 9y - 7z = -8

This guide demonstrates how to solve a system of three linear equations: x + y - z = -2, -3x + 9y - 5z = 6, and x + 9y - 7z = -8. Explore step-by-step the elimination method used to find the values of x, y, and z. Ideal for students in Grades 9-11 learning algebra and systems of equations.

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