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To determine if \((-1, 1, -2)\) satisfies each equation in the system, we’ll substitute \( x_1 = -1 \), \( x_2 = 1 \), and \( x_3 = -2 \) into each equation and check if they hold true.

The system of equations is:

1. \(5x_1 + 2x_2 + x_3 = -5\)
2. \(6x_1 + 3x_2 + x_3 = -5\)
3. \(4x_1 + 2x_2 + x_3 = -3\)

### Step-by-Step Substitution

#### Equation 1
\[
5(-1) + 2(1) + (-2) = -5
\]
Calculating each term:
\[
-5 + 2 - 2 = -5
\]
The first equation is satisfied.

#### Equation 2
\[
6(-1) + 3(1) + (-2) = -5
\]
Calculating each term:
\[
-6 + 3 - 2 = -5
\]
The second equation is also satisfied.

#### Equation 3
\[
4(-1) + 2(1) + (-2) = -3
\]
Calculating each term:
\[
-4 + 2 - 2 = -4
\]
The third equation is **not** satisfied, as \(-4 \neq -3\).

### Conclusion
Since \((-1, 1, -2)\) does not satisfy all three equations (specifically the third equation), it is **not** a solution to the system.

Would you like a more detailed breakdown or have any questions on this solution?

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Here are some related questions for further practice:
1. How do you determine if a point satisfies a system of equations?
2. What methods can solve a system of linear equations?
3. Can a point partially satisfy a system of equations?
4. What is the geometric interpretation of a solution to a system of equations?
5. How can you verify a solution using matrix operations?

**Tip:** When substituting values into equations, double-check each arithmetic operation to ensure accuracy.

Determine whether (-1, 1, -2) satisfies each equation in the following system of equations:

1. 5x1 + 2x2 + x3 = -5
2. 6x1 + 3x2 + x3 = -5
3. 4x1 + 2x2 + x3 = -3

Is (-1, 1, -2) a solution of the system of equations?

Learn how to determine if the point (-1, 1, -2) satisfies each equation in a given system of linear equations. This problem provides a step-by-step substitution method, ideal for students in Grades 10-12 studying algebra.

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