what is the x,y, and z if
x+y+z=-2
x-y+z=8
x+2y-z=-17

We are given the following system of equations:

\[
1. \ x + y + z = -2
\]
\[
2. \ x - y + z = 8
\]
\[
3. \ x + 2y - z = -17
\]

We will solve this system using elimination or substitution to find the values of \(x\), \(y\), and \(z\).

### Step 1: Subtract Equation 1 from Equation 2
\[
(x - y + z) - (x + y + z) = 8 - (-2)
\]
This simplifies to:
\[
-2y = 10 \implies y = -5
\]

### Step 2: Substitute \(y = -5\) into Equation 1
\[
x + (-5) + z = -2
\]
This simplifies to:
\[
x + z = 3 \tag{4}
\]

### Step 3: Substitute \(y = -5\) into Equation 3
\[
x + 2(-5) - z = -17
\]
This simplifies to:
\[
x - 10 - z = -17 \implies x - z = -7 \tag{5}
\]

### Step 4: Solve Equations 4 and 5
We now have the system:
\[
x + z = 3 \tag{4}
\]
\[
x - z = -7 \tag{5}
\]

Add these two equations:
\[
(x + z) + (x - z) = 3 + (-7)
\]
This simplifies to:
\[
2x = -4 \implies x = -2
\]

### Step 5: Substitute \(x = -2\) into Equation 4
\[
-2 + z = 3 \implies z = 5
\]

### Final Solution:
\[
x = -2, \ y = -5, \ z = 5
\]

Would you like further details on the steps? 

Here are five related questions you might find helpful:
1. How do we solve systems of equations using substitution?
2. What is the elimination method in solving systems of equations?
3. Can we solve systems of equations using matrices?
4. How can graphical methods be used to solve systems of equations?
5. What is the role of determinants in solving linear equations?

**Tip**: Always double-check your solution by substituting the values back into the original equations.

what is the x, y, and z if x+y+z=-2, x-y+z=8, x+2y-z=-17

Learn how to solve the system of equations x + y + z = -2, x - y + z = 8, and x + 2y - z = -17 using substitution and elimination methods. This problem covers key algebraic concepts and methods for solving linear systems, suitable for students in Grades 8-10.

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