We have a system of three linear equations:

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

### Step 1: Simplify the System
Let's simplify the system by solving for one variable and substituting it into the other equations.

1. From Equation 1, solve for \(x\):
   \[
   x = -1 - 2y
   \]
   Substitute this expression for \(x\) into Equations 2 and 3.

### Step 2: Substitution into Equations 2 and 3
Substitute \(x = -1 - 2y\) into Equation 2:
\[
-(-1 - 2y) + 3y + 2z = -4
\]
Simplify:
\[
1 + 2y + 3y + 2z = -4
\]
\[
5y + 2z = -5 \quad \text{(Equation 4)}
\]

Substitute \(x = -1 - 2y\) into Equation 3:
\[
-(-1 - 2y) + y - 4z = 10
\]
Simplify:
\[
1 + 2y + y - 4z = 10
\]
\[
3y - 4z = 9 \quad \text{(Equation 5)}
\]

### Step 3: Solve the Reduced System
Now, solve the system formed by Equations 4 and 5:
\[
\begin{aligned}
4) & \quad 5y + 2z = -5 \\
5) & \quad 3y - 4z = 9
\end{aligned}
\]

Multiply Equation 4 by 2 and Equation 5 by 1 to eliminate \(z\):

\[
10y + 4z = -10 \quad \text{(Equation 6)}
\]
\[
3y - 4z = 9 \quad \text{(Equation 5)}
\]

### Step 4: Add Equations 6 and 5
Add Equation 6 to Equation 5 to eliminate \(z\):
\[
(10y + 4z) + (3y - 4z) = -10 + 9
\]
Simplify:
\[
13y = -1
\]
\[
y = -\frac{1}{13}
\]

### Step 5: Solve for \(z\)
Substitute \(y = -\frac{1}{13}\) into Equation 4:
\[
5\left(-\frac{1}{13}\right) + 2z = -5
\]
Simplify:
\[
-\frac{5}{13} + 2z = -5
\]
Multiply through by 13:
\[
-5 + 26z = -65
\]
\[
26z = -60
\]
\[
z = -\frac{30}{13}
\]

### Step 6: Solve for \(x\)
Now, substitute \(y = -\frac{1}{13}\) back into the expression for \(x\):
\[
x = -1 - 2\left(-\frac{1}{13}\right)
\]
Simplify:
\[
x = -1 + \frac{2}{13}
\]
\[
x = -\frac{13}{13} + \frac{2}{13} = -\frac{11}{13}
\]

### Final Solution
\[
x = -\frac{11}{13}, \quad y = -\frac{1}{13}, \quad z = -\frac{30}{13}
\]

Would you like further details or have any questions?

### Related Questions
1. How do you solve a system of linear equations using substitution?
2. What is the significance of the determinant in solving systems of linear equations?
3. Can you explain the method of elimination in solving linear equations?
4. What are some practical applications of solving systems of linear equations?
5. How does one solve a system of linear equations with three variables graphically?

### Tip
Always double-check your calculations by substituting your solutions back into the original equations to verify correctness.

Learn how to solve a system of three linear equations with three variables using the substitution and elimination methods. This detailed step-by-step solution covers algebraic concepts and methods crucial for high school students.

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