Find the solution to the system of equations:
⎧⎪⎨⎪⎩x−2y+z=9y+2z=−10x+y+3z=−9{x-2y+z=9y+2z=-10x+y+3z=-9


(x,y,z)=

We are given the system of equations:

\[
\begin{aligned}
1) \quad & x - 2y + z = 9 \\
2) \quad & y + 2z = -10 \\
3) \quad & x + y + 3z = -9
\end{aligned}
\]

Let's solve this step-by-step.

### Step 1: Solve for \( y \) from Equation (2)

From Equation (2), we can solve for \( y \) in terms of \( z \):

\[
y = -10 - 2z
\]

### Step 2: Substitute \( y = -10 - 2z \) into Equations (1) and (3)

Now, substitute \( y = -10 - 2z \) into Equations (1) and (3).

#### Substitute into Equation (1):
\[
x - 2(-10 - 2z) + z = 9
\]
\[
x + 20 + 4z + z = 9
\]
\[
x + 5z = -11 \quad \text{(Equation A)}
\]

#### Substitute into Equation (3):
\[
x + (-10 - 2z) + 3z = -9
\]
\[
x - 10 - 2z + 3z = -9
\]
\[
x + z = 1 \quad \text{(Equation B)}
\]

### Step 3: Solve the system of two equations in \( x \) and \( z \)

Now we have the system:
1) \( x + 5z = -11 \)
2) \( x + z = 1 \)

Subtract Equation B from Equation A:
\[
(x + 5z) - (x + z) = -11 - 1
\]
\[
4z = -12
\]
\[
z = -3
\]

### Step 4: Substitute \( z = -3 \) back into Equation B

Substitute \( z = -3 \) into Equation B:
\[
x + (-3) = 1
\]
\[
x = 4
\]

### Step 5: Substitute \( z = -3 \) into the expression for \( y \)

Now, substitute \( z = -3 \) into \( y = -10 - 2z \):
\[
y = -10 - 2(-3) = -10 + 6 = -4
\]

### Final Solution

The solution to the system of equations is:
\[
(x, y, z) = (4, -4, -3)
\]

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

Here are 5 related questions you can try:
1. Solve the system: \( x + y + z = 3 \), \( 2x - y + z = 5 \), \( 3x + 4y - 2z = -2 \).
2. Solve the system: \( 2x + y - z = 8 \), \( x - 2y + 3z = -6 \), \( 3x + 3y + z = 12 \).
3. Solve a 3-variable system using matrix methods (Gaussian elimination).
4. What happens to the system if all equations are multiplied by a scalar?
5. How do we solve non-linear systems of equations?

**Tip**: When solving systems of equations, substituting one variable into another is an efficient strategy to reduce the number of variables step by step.

Find the solution to the system of equations: ⎧⎪⎨⎪⎩ x − 2y + z = 9 y + 2z = −10 x + y + 3z = −9

Learn how to solve the system of equations x - 2y + z = 9, y + 2z = -10, and x + y + 3z = -9 with a step-by-step solution using the substitution method. This problem explores algebraic concepts and the theory behind solving linear systems, suitable for students in Grades 9-11.

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