Math Problem Statement

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.

Solution

The system of equations provided in the image is:

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.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Least Squares Method
Over-determined Systems

Formulas

A * x = b (Matrix Form of a System of Equations)
Normal Equation: A^T * A * x = A^T * b

Theorems

Normal Equation Theorem in Least Squares Approximation

Suitable Grade Level

Undergraduate (Linear Algebra)

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