We are given the following system of equations:

\[
1. \ 10x + y + 2y = 8 \quad \text{(simplifies to } 10x + 3y = 8\text{)}
\]
\[
2. \ -x - y + z = 10
\]
\[
3. \ 4x - y + z = 5
\]

### Step 1: Simplify the first equation

\[
10x + y + 2y = 8 \quad \Rightarrow \quad 10x + 3y = 8
\]

Now the system becomes:

\[
1. \ 10x + 3y = 8
\]
\[
2. \ -x - y + z = 10
\]
\[
3. \ 4x - y + z = 5
\]

### Step 2: Eliminate \( z \) by subtracting equation 2 from equation 3

\[
(4x - y + z) - (-x - y + z) = 5 - 10
\]
\[
4x - y + z + x + y - z = -5
\]
\[
5x = -5
\]
\[
x = -1
\]

### Step 3: Substitute \( x = -1 \) into equations 1 and 2

#### Substituting into equation 1:
\[
10(-1) + 3y = 8
\]
\[
-10 + 3y = 8 \quad \Rightarrow \quad 3y = 18 \quad \Rightarrow \quad y = 6
\]

#### Substituting into equation 2:
\[
-(-1) - 6 + z = 10
\]
\[
1 - 6 + z = 10 \quad \Rightarrow \quad -5 + z = 10 \quad \Rightarrow \quad z = 15
\]

### Final solution:
\[
x = -1, \quad y = 6, \quad z = 15
\]

Would you like a more detailed explanation, or do you have any questions?

Here are 5 related questions you can explore:
1. How can you solve a system of three equations with three unknowns using matrix methods?
2. What are the different methods to solve a system of linear equations (e.g., substitution, elimination, etc.)?
3. How do you solve a system of equations with inconsistent or dependent equations?
4. Can you always expect a unique solution to a system of linear equations?
5. How can graphical methods be used to solve a system of linear equations?

**Tip:** When solving systems of equations, always look for opportunities to eliminate one variable early on to simplify the process.

10x + y + 2y = 8, -x - y + z = 10, 4x - y + z = 5

Learn how to solve a system of linear equations step by step: 10x + 3y = 8, -x - y + z = 10, 4x - y + z = 5. This detailed explanation uses elimination and substitution methods to find x, y, and z. Perfect for students in Grades 10-12.

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