https://file.aiquickdraw.com/mathbot/file/1729084228577olmt41o4.jpg

The system of equations given is:

\[
\begin{aligned}
1) & \quad x + 9y + z = 20 \\
2) & \quad x + 10y - 2z = 18 \\
3) & \quad 3x + 27y + 2z = 58 \\
\end{aligned}
\]

### Step 1: Solve using elimination or substitution
We can use elimination to solve this system.

#### Subtract Equation 1 from Equation 2:
\[
(x + 10y - 2z) - (x + 9y + z) = 18 - 20
\]
\[
x + 10y - 2z - x - 9y - z = -2
\]
\[
y - 3z = -2 \quad \text{(Equation 4)}
\]

#### Subtract 3 times Equation 1 from Equation 3:
\[
(3x + 27y + 2z) - 3(x + 9y + z) = 58 - 3(20)
\]
\[
3x + 27y + 2z - 3x - 27y - 3z = 58 - 60
\]
\[
-z = -2 \quad \Rightarrow \quad z = 2
\]

### Step 2: Substitute \( z = 2 \) into Equation 4:
\[
y - 3(2) = -2
\]
\[
y - 6 = -2 \quad \Rightarrow \quad y = 4
\]

### Step 3: Substitute \( y = 4 \) and \( z = 2 \) into Equation 1:
\[
x + 9(4) + 2 = 20
\]
\[
x + 36 + 2 = 20 \quad \Rightarrow \quad x + 38 = 20 \quad \Rightarrow \quad x = -18
\]

### Final Solution:
The solution as an ordered triple is:
\[
(x, y, z) = (-18, 4, 2)
\]

Let me know if you'd like more details or have any further questions!

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Here are some related questions for further practice:
1. How can you verify if the solution \((-18, 4, 2)\) satisfies all the equations?
2. What is the geometric interpretation of a system of linear equations?
3. How does substitution compare to elimination when solving a system of equations?
4. What are some special cases that can arise when solving a system of equations (e.g., no solution, infinite solutions)?
5. How can you use matrices to solve this system?

**Tip:** Always check your solution by substituting it back into all original equations to ensure correctness!

Solve the system of equations:
1) x + 9y + z = 20
2) x + 10y - 2z = 18
3) 3x + 27y + 2z = 58

Learn how to solve a system of three linear equations using elimination and substitution. This problem involves key algebraic concepts, including the manipulation of linear equations, substitution, and elimination. Suitable for students in Grades 9-12.

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