Solve the system using the elimination method. x+y-2z=5, -x+2y+z=2, 2x+3y-z=9

To solve the system of equations using the elimination method, we have the following equations:

\[
\begin{aligned}
1) & \quad x + y - 2z = 5 \\
2) & \quad -x + 2y + z = 2 \\
3) & \quad 2x + 3y - z = 9 \\
\end{aligned}
\]

### Step 1: Eliminate one variable
We'll start by eliminating \( x \) from equations 1) and 2). We can add these two equations to eliminate \( x \):

\[
(x + y - 2z) + (-x + 2y + z) = 5 + 2
\]

Simplifying this:

\[
y + 2y - 2z + z = 7 \\
3y - z = 7 \quad \text{(Equation 4)}
\]

Next, let's eliminate \( x \) from equations 2) and 3). Add equation 2) to twice equation 3):

\[
(-x + 2y + z) + 2(2x + 3y - z) = 2 + 2(9)
\]

Expanding and simplifying:

\[
- x + 2y + z + 4x + 6y - 2z = 20 \\
3x + 8y - z = 20 \quad \text{(Equation 5)}
\]

### Step 2: Eliminate another variable
Now, we'll use equations 4) and 5) to eliminate \( z \). We can subtract equation 4) from equation 5) after multiplying equation 4) by 1 to align the coefficients of \( z \):

\[
(3x + 8y - z) - (3y - z) = 20 - 7
\]

Simplifying:

\[
3x + 5y = 13 \quad \text{(Equation 6)}
\]

### Step 3: Solve for \( y \) and \( x \)
Now, we solve equation 6) for \( y \) and then substitute back to find \( x \).

\[
y = \frac{13 - 3x}{5}
\]

Substitute \( y \) back into equation 4):

\[
3\left(\frac{13 - 3x}{5}\right) - z = 7
\]

Solve for \( z \), and then substitute all values into one of the original equations to find \( x \).

### Step 4: Back-substitute to find \( z \)
Let's simplify this and solve the system:

1. Solve for \( y \) from equation 6): \( y = \frac{13 - 3x}{5} \)
2. Substitute \( y \) into equation 4) to find \( z \).
3. Substitute \( y \) and \( z \) into one of the original equations to find \( x \).

Finally, after solving:

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

So, the solution to the system is:

\[
(x, y, z) = (2, 1, -2)
\]

Would you like more details or have any questions about this solution?

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Here are 5 related questions:
1. Can you explain how to eliminate variables in a different system of equations?
2. How would you solve the system using the substitution method instead?
3. What is the determinant of the coefficient matrix for this system, and what does it signify?
4. How would the solution change if the constant terms were different?
5. Can the elimination method be used for systems with more than three variables?

**Tip:** Always check your solution by substituting the values back into the original equations to ensure they satisfy all equations.

Learn how to solve a system of linear equations using the elimination method. Step-by-step solution to the equations x+y-2z=5, -x+2y+z=2, 2x+3y-z=9, covering concepts of elimination, substitution, and verification of solutions. Suitable for Grades 10-12.

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