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We are given the system \(Ax = b\), where:

\[
A = \begin{bmatrix} -1 & 3 \\ -4 & 12 \end{bmatrix}, \quad b = \begin{bmatrix} 1 \\ -1 \end{bmatrix}
\]
A basis for the null space of \(A\) is given by:
\[
\begin{bmatrix} 3 \\ 1 \end{bmatrix}
\]
and a particular solution to \(Ax = b\) is:
\[
x_p = \begin{bmatrix} -1 \\ 0 \end{bmatrix}
\]

We need to find the row space solution \(x_r\) for the equation \(Ax = b\).

### Solution

The general solution for the equation \(Ax = b\) can be written as:
\[
x = x_p + x_n
\]
where \(x_p\) is a particular solution to the system, and \(x_n\) is any solution in the null space of \(A\).

Given that a basis for the null space is \(\begin{bmatrix} 3 \\ 1 \end{bmatrix}\), we can express \(x_n\) as:
\[
x_n = k \begin{bmatrix} 3 \\ 1 \end{bmatrix}
\]
for some scalar \(k\).

The row space solution \(x_r\) represents the component of \(x\) that lies in the row space of \(A\). Since \(x_p = \begin{bmatrix} -1 \\ 0 \end{bmatrix}\) is already a solution to the system, and it does not lie in the null space of \(A\), it is the row space solution.

Therefore, the row space solution \(x_r\) is:
\[
x_r = \begin{bmatrix} -1 \\ 0 \end{bmatrix}
\]

Would you like more details on this explanation or have any questions?

### Related Questions
1. What is the definition of the row space of a matrix?
2. How do you find a particular solution for a linear system \(Ax = b\)?
3. What are the steps to find the null space of a matrix?
4. Why is the null space important when solving linear equations?
5. How can we verify if a given vector is a solution to the equation \(Ax = b\)?

### Tip
When solving a system of linear equations, it's helpful to check both the particular solution and the null space solution to ensure the completeness of the general solution.

Consider the system Ax = b where A = [[-1, 3], [-4, 12]] and b = [[1], [-1]]. A basis for the null space of A is given by [[3], [1]] and x = [[-1], [0]] is a particular solution. Use this information to find the row space solution x_r to Ax = b.

This problem involves solving a system of linear equations Ax = b where A = [[-1, 3], [-4, 12]] and b = [[1], [-1]]. A particular solution and null space basis are provided, and the task is to find the row space solution for Ax = b. Learn about key linear algebra concepts like row space, null space, and general solutions to systems of equations.

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