Math Problem Statement

solve the problem in depth

Find any x, y, z that gives the 'closest' possible answer to the inconsistent linear system of equations: 1. x + y - z = 7 2. x + 2y - 7z = -2 3. 2x + 3y - 8z = -7 4. 3x + 2y + 3z = 3

Solution

The problem involves solving an inconsistent system of linear equations by finding the "closest" possible solution. To do so, we use the least-squares approximation method, which minimizes the error between the given equations.

The system of equations is:

  1. x+y−z=7x + y - z = 7
  2. x+2y−7z=−2x + 2y - 7z = -2
  3. 2x+3y−8z=−72x + 3y - 8z = -7
  4. 3x+2y+3z=33x + 2y + 3z = 3

Let me outline the process to solve this:

Step 1: Write the system in matrix form

The system can be expressed as: Av=bA \mathbf{v} = \mathbf{b} where:

\begin{bmatrix} 1 & 1 & -1 \\ 1 & 2 & -7 \\ 2 & 3 & -8 \\ 3 & 2 & 3 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 7 \\ -2 \\ -7 \\ 3 \end{bmatrix}.$$ ### Step 2: Least-squares approximation Since the system is inconsistent, we approximate the solution by minimizing the error: $$\| A \mathbf{v} - \mathbf{b} \|^2$$ This is achieved by solving the **normal equations**: $$A^\top A \mathbf{v} = A^\top \mathbf{b}.$$ ### Step 3: Compute $$A^\top A$$ and $$A^\top \mathbf{b}$$ $$A^\top = \begin{bmatrix} 1 & 1 & 2 & 3 \\ 1 & 2 & 3 & 2 \\ -1 & -7 & -8 & 3 \end{bmatrix}.$$ $$A^\top A = \begin{bmatrix} 15 & 17 & -17 \\ 17 & 18 & -33 \\ -17 & -33 & 123 \end{bmatrix}.$$ $$A^\top \mathbf{b} = \begin{bmatrix} 8 \\ 2 \\ -91 \end{bmatrix}.$$ ### Step 4: Solve $$A^\top A \mathbf{v} = A^\top \mathbf{b}$$ The equation to solve is: $$\begin{bmatrix} 15 & 17 & -17 \\ 17 & 18 & -33 \\ -17 & -33 & 123 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 8 \\ 2 \\ -91 \end{bmatrix}.$$ Using numerical computation, let me solve for $$x, y, z$$. numpy as np # Define the matrices A^T A and A^T b ATA = np.array([[15, 17, -17], [17, 18, -33], [-17, -33, 123]]) ATb = np.array([8, 2, -91]) # Solve the system ATA * v = ATb solution = np.linalg.solve(ATA, ATb) solutionThe least-squares solution to the system is approximately: $$x = -6.072, \quad y = 5.807, \quad z = -0.021.$$ ### Explanation This solution minimizes the discrepancy between the left-hand side and right-hand side of the original inconsistent system. While the equations cannot all be satisfied simultaneously, this solution represents the "closest" values for $$x$$, $$y$$, and $$z$$ that fit the system in the least-squares sense. Would you like a step-by-step verification or further explanation? Here are some follow-up questions you might consider: 1. How does the least-squares method work conceptually in linear algebra? 2. Why is the system of equations inconsistent, and how can you verify this? 3. What is the geometric interpretation of the least-squares solution? 4. Can we validate the solution by calculating the residuals? 5. What are other methods to handle inconsistent systems? **Tip:** Always check the residuals $$A\mathbf{v} - \mathbf{b}$$ to understand how well the solution approximates the system!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Operations
Least-Squares Approximation
Inconsistent Systems

Formulas

Least-squares solution: AᵀA * x = Aᵀb
Residual minimization: ||A * v - b||²

Theorems

Normal Equations for Least-Squares Approximation

Suitable Grade Level

Undergraduate (Linear Algebra)

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