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We are given a system of three equations:

\[
\begin{aligned}
1) & \ 10x + y + 2z = 8 \\
2) & \ -x - y + z = 10 \\
3) & \ 4x - y + z = 5
\end{aligned}
\]

We will solve this system step by step using the substitution or elimination method.

### Step 1: Solve one equation for one variable
Let’s solve the second equation for \(x\) in terms of \(y\) and \(z\).

From equation 2:

\[
-x - y + z = 10 \implies x = -y + z - 10
\]

### Step 2: Substitute into the other equations
Now, substitute \(x = -y + z - 10\) into equations 1 and 3.

**Substitute into equation 1:**
\[
10(-y + z - 10) + y + 2z = 8
\]
Simplifying:
\[
-10y + 10z - 100 + y + 2z = 8
\]
\[
-9y + 12z = 108
\]
\[
9y - 12z = -108 \quad \text{(Equation 4)}
\]

**Substitute into equation 3:**
\[
4(-y + z - 10) - y + z = 5
\]
Simplifying:
\[
-4y + 4z - 40 - y + z = 5
\]
\[
-5y + 5z = 45
\]
\[
y - z = -9 \quad \text{(Equation 5)}
\]

### Step 3: Solve the system of two equations
Now we have the system:
\[
9y - 12z = -108 \quad \text{(Equation 4)}
\]
\[
y - z = -9 \quad \text{(Equation 5)}
\]

Solve equation 5 for \(y\):
\[
y = z - 9
\]

Substitute \(y = z - 9\) into equation 4:
\[
9(z - 9) - 12z = -108
\]
Simplify:
\[
9z - 81 - 12z = -108
\]
\[
-3z - 81 = -108
\]
\[
-3z = -27
\]
\[
z = 9
\]

### Step 4: Solve for \(y\) and \(x\)
Substitute \(z = 9\) into \(y = z - 9\):
\[
y = 9 - 9 = 0
\]

Now substitute \(y = 0\) and \(z = 9\) into \(x = -y + z - 10\):
\[
x = -0 + 9 - 10 = -1
\]

### Final Solution:
The solution to the system is:
\[
x = -1, \quad y = 0, \quad z = 9
\]

Let me know if you need further details or have any questions! Here are some related questions for further practice:

1. How can you verify that the solution satisfies all three original equations?
2. What is the graphical interpretation of solving a system of three equations?
3. How would the solution change if the right-hand side constants were different?
4. Can the system have no solution or infinitely many solutions? If so, how can we identify those cases?
5. How would you solve the system using matrix methods, such as Gaussian elimination?

**Tip:** Always substitute your solution back into the original equations to verify correctness!

Solve the following system of equations for all three variables.

1) 10x + y + 2z = 8
2) -x - y + z = 10
3) 4x - y + z = 5

Learn how to solve the system of linear equations: 10x + y + 2z = 8, -x - y + z = 10, and 4x - y + z = 5. This detailed solution uses substitution and elimination methods to find the values of x, y, and z. Ideal for students in Grades 9-12.

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