The least squares method is used to find the best approximate solution to a system of equations that may not have an exact solution. The idea is to minimize the sum of the squares of the residuals (differences between the actual and predicted values).

Given the system of equations:

1. $3x_1 - 2x_2 = 4$
2. $x_1 + x_2 = 2$
3. $x_1 - x_2 = 1$

We will find the least squares solution as follows:

Step 1: Represent the system in matrix form

The system can be represented as:

$Ax = b$

where:

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

Step 2: Compute the least squares solution

The least squares solution is given by the formula:

$x = (A^T A)^{-1} A^T b$

Let's calculate this step-by-step.

1. Compute $A^T$, the transpose of $A$.
2. Compute $A^T A$.
3. Compute $(A^T A)^{-1}$, the inverse of $A^T A$.
4. Compute $A^T b$.
5. Multiply $(A^T A)^{-1}$ by $A^T b$ to get $x$.

I'll perform these calculations now.The least squares solution to the given system of equations is approximately:

$x_1 \approx 1.6, \quad x_2 \approx 0.433$

This solution minimizes the sum of the squared residuals of the system.

Would you like more details on any step, or do you have any questions?

Related Questions

1. How is the least squares method used in data fitting or regression analysis?
2. What are the properties of the matrix $(A^T A)$ in the least squares method?
3. Can the least squares method be applied to non-linear systems?
4. What are the implications of a non-invertible $(A^T A)$ matrix in the least squares method?
5. How can we interpret the residuals in the context of least squares solutions?

Tip

For systems where there are more equations than variables, the least squares method is particularly useful as it provides an approximate solution by minimizing the error.