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We are asked to express the vector \(\mathbf{v} = (-4, 7, 1)\) as a linear combination of the vectors \(\mathbf{v_1} = (0, 1, 1)\), \(\mathbf{v_2} = (1, 2, -2)\), and \(\mathbf{v_3} = (3, 1, 0)\).

In other words, we need to find scalars \(a\), \(b\), and \(c\) such that:
\[
\mathbf{v} = a \mathbf{v_1} + b \mathbf{v_2} + c \mathbf{v_3}
\]
This can be written as the following system of equations based on the components of each vector:

\[
a(0, 1, 1) + b(1, 2, -2) + c(3, 1, 0) = (-4, 7, 1)
\]

Breaking it down into three separate equations for each component:

1. For the first component:
\[
0a + 1b + 3c = -4
\]
\[
b + 3c = -4
\]

2. For the second component:
\[
1a + 2b + 1c = 7
\]
\[
a + 2b + c = 7
\]

3. For the third component:
\[
1a - 2b + 0c = 1
\]
\[
a - 2b = 1
\]

### Solving the system of equations:

We have the following system of three equations:
1. \( b + 3c = -4 \)
2. \( a + 2b + c = 7 \)
3. \( a - 2b = 1 \)

Let’s solve this step by step.

1. Solve for \(b\) in terms of \(c\) from equation 1:
\[
b = -4 - 3c
\]

2. Substitute this expression for \(b\) into equation 2:
\[
a + 2(-4 - 3c) + c = 7
\]
\[
a - 8 - 6c + c = 7
\]
\[
a - 8 - 5c = 7
\]
\[
a = 15 + 5c
\]

3. Substitute this expression for \(a\) and \(b = -4 - 3c\) into equation 3:
\[
(15 + 5c) - 2(-4 - 3c) = 1
\]
\[
15 + 5c + 8 + 6c = 1
\]
\[
23 + 11c = 1
\]
\[
11c = -22
\]
\[
c = -2
\]

4. Substitute \(c = -2\) into the equation for \(b\):
\[
b = -4 - 3(-2) = -4 + 6 = 2
\]

5. Finally, substitute \(c = -2\) into the equation for \(a\):
\[
a = 15 + 5(-2) = 15 - 10 = 5
\]

### Final Answer:
The scalars are:
\[
a = 5, \quad b = 2, \quad c = -2
\]

Thus, the vector \(\mathbf{v} = (-4, 7, 1)\) can be written as:
\[
\mathbf{v} = 5\mathbf{v_1} + 2\mathbf{v_2} - 2\mathbf{v_3}
\]

Would you like a more detailed breakdown of any step? 

Here are five related questions:
1. How can we solve a system of linear equations using matrices?
2. What is the geometric interpretation of expressing vectors as linear combinations?
3. How do we know if a set of vectors forms a basis in \(\mathbb{R}^3\)?
4. What is the condition for three vectors to be linearly independent in \(\mathbb{R}^3\)?
5. How can linear transformations be represented using matrix operations?

**Tip**: When solving systems of equations, substitution works well when one variable is easy to isolate. For larger systems, consider matrix methods like Gaussian elimination.

Express v = (-4, 7, 1) in R^3 as a linear combination of the vectors v1 = (0, 1, 1), v2 = (1, 2, -2), v3 = (3, 1, 0).

Solve the problem of expressing the vector v = (-4, 7, 1) as a linear combination of the vectors v1 = (0, 1, 1), v2 = (1, 2, -2), and v3 = (3, 1, 0) in R^3. Learn how to use systems of equations to find the solution.

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