https://file.aiquickdraw.com/mathbot/file/1726896077308rv49c1cf.jpg

The system of equations provided in the image is:

\[
\begin{aligned}
    3x_1 - 2x_2 &= 4 \\
    x_1 + x_2 &= 2 \\
    x_1 - x_2 &= 1
\end{aligned}
\]

To find the best approximate solution using the **least squares method**, we must first represent the system of equations in matrix form as \( A \mathbf{x} = \mathbf{b} \), where:

\[
A = \begin{bmatrix} 3 & -2 \\ 1 & 1 \\ 1 & -1 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 4 \\ 2 \\ 1 \end{bmatrix}
\]

In the least squares method, the goal is to minimize the error between the actual and predicted values, which is done by solving the normal equation:

\[
A^T A \mathbf{x} = A^T \mathbf{b}
\]

I'll compute this step-by-step to find the best approximate solution for \( \mathbf{x} \).The best approximate solution to the system of equations using the least squares method is:

\[
x_1 \approx 1.6, \quad x_2 \approx 0.433
\]

Would you like a detailed explanation of the steps involved in solving this problem, or do you have any other questions?

Here are some related questions you might consider:
1. How is the least squares method used to approximate solutions in over-determined systems?
2. What is the significance of the normal equation \( A^T A \mathbf{x} = A^T \mathbf{b} \)?
3. In which cases do we prefer least squares approximation instead of finding an exact solution?
4. Can the least squares method be extended to non-linear systems?
5. How would you calculate the error or residual for this least squares solution?

**Tip:** The least squares method is particularly useful when a system of equations has no exact solution, such as when the system is over-determined.

Using the least squares method, find the best approximate solution to the following system of equations: 3x1 - 2x2 = 4, x1 + x2 = 2, x1 - x2 = 1.

Find the best approximate solution for the system of equations using the least squares method. This problem covers key concepts like linear algebra, matrix representation, and solving over-determined systems. Suitable for undergraduate students learning the least squares method.

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