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The task asks us to verify that the given parametric vector \((1 - t, 2 + 3t, 3 - 2t)\) is a solution to the given system of linear equations for all values of \( t \). Let's break this down step by step.

### Given:

The vector is:
\[
(x_1, x_2, x_3) = (1 - t, 2 + 3t, 3 - 2t)
\]
The system of linear equations is:
\[
\begin{aligned}
x_1 + x_2 + x_3 &= 6 \\
x_1 - x_2 - 2x_3 &= -7 \\
5x_1 + x_2 - x_3 &= 4
\end{aligned}
\]

### Substituting the parametric expressions for \( x_1 \), \( x_2 \), and \( x_3 \):

1. **First equation**: 
   \[
   x_1 + x_2 + x_3 = 6
   \]
   Substituting \( x_1 = 1 - t \), \( x_2 = 2 + 3t \), and \( x_3 = 3 - 2t \):
   \[
   (1 - t) + (2 + 3t) + (3 - 2t) = 6
   \]
   Simplifying:
   \[
   1 - t + 2 + 3t + 3 - 2t = 6
   \]
   \[
   6 + 0t = 6
   \]
   This is true for all values of \( t \).

2. **Second equation**:
   \[
   x_1 - x_2 - 2x_3 = -7
   \]
   Substituting \( x_1 = 1 - t \), \( x_2 = 2 + 3t \), and \( x_3 = 3 - 2t \):
   \[
   (1 - t) - (2 + 3t) - 2(3 - 2t) = -7
   \]
   Simplifying:
   \[
   1 - t - 2 - 3t - 6 + 4t = -7
   \]
   \[
   -7 + 0t = -7
   \]
   This is also true for all values of \( t \).

3. **Third equation**:
   \[
   5x_1 + x_2 - x_3 = 4
   \]
   Substituting \( x_1 = 1 - t \), \( x_2 = 2 + 3t \), and \( x_3 = 3 - 2t \):
   \[
   5(1 - t) + (2 + 3t) - (3 - 2t) = 4
   \]
   Simplifying:
   \[
   5 - 5t + 2 + 3t - 3 + 2t = 4
   \]
   \[
   4 + 0t = 4
   \]
   This is true for all values of \( t \).

### Conclusion:
The vector \((1 - t, 2 + 3t, 3 - 2t)\) satisfies all three equations for all values of \( t \), hence it is a solution to the system for any \( t \).

Would you like more details or have any questions?

Here are 5 related questions to expand on this:
1. What is the general method to solve systems of linear equations using substitution?
2. How can we interpret parametric solutions geometrically?
3. What are some other methods to solve linear systems besides substitution?
4. Can parametric solutions apply to non-linear systems?
5. How does Gaussian elimination work in solving systems of equations?

**Tip:** When dealing with systems of linear equations, always check if the number of equations matches the number of variables to ensure a well-posed problem!

Verify that for all values of t, (1 - t, 2 + 3t, 3 - 2t) is a solution to the linear system x1 + x2 + x3 = 6, x1 - x2 - 2x3 = -7, 5x1 + x2 - x3 = 4.

Learn how to verify that the parametric solution (1 - t, 2 + 3t, 3 - 2t) solves the given system of linear equations for all values of t. Step-by-step explanations of substitution and verification for algebra students in Grades 10-12.

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