Question

Following observation equations are obtained in a survey task.
\[ x + y = 3 \\ 2x + y = 6 \\ x + 2y = 4 \] Using least square method, the most probable values of x and y will be

• A. \( x = 2.10, y = 0.90 \)
• B. \( x = 2.64, y = 0.64 \)
• C. \( x = 2.51, y = 0.51 \)
• D. \( x = 2.75, y = 0.75 \)

Hint:

In the least square method, when solving a system of over-determined equations, calculate the normal equation \( \mathbf{A}^T \mathbf{A} \mathbf{X} = \mathbf{A}^T \mathbf{b} \) and solve for the unknowns.

Answer 1

The Correct Option is B

The least square method is used to solve over-determined systems of equations. The given system of equations is: \[ x + y = 3 \quad \text{(Equation 1)} \\ 2x + y = 6 \quad \text{(Equation 2)} \\ x + 2y = 4 \quad \text{(Equation 3)} \] We can write these equations in matrix form: \[ \begin{bmatrix} 1 & 1 \\ 2 & 1 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 3 \\ 6 \\ 4 \end{bmatrix} \] To solve this using the least square method, we calculate the normal equation: \[ \mathbf{A}^T \mathbf{A} \mathbf{X} = \mathbf{A}^T \mathbf{b} \] Where: \[ \mathbf{A} = \begin{bmatrix} 1 & 1 \\ 2 & 1 \\ 1 & 2 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 3 \\ 6 \\ 4 \end{bmatrix} \] First, calculate \( \mathbf{A}^T \mathbf{A} \): \[ \mathbf{A}^T \mathbf{A} = \begin{bmatrix} 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix} \begin{bmatrix} 1 & 1 \\ 2 & 1 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 6 & 4 \\ 4 & 6 \end{bmatrix} \] Next, calculate \( \mathbf{A}^T \mathbf{b} \): \[ \mathbf{A}^T \mathbf{b} = \begin{bmatrix} 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix} \begin{bmatrix} 3 \\ 6 \\ 4 \end{bmatrix} = \begin{bmatrix} 18 \\ 18 \end{bmatrix} \] Now, solve for \( \mathbf{X} \): \[ \begin{bmatrix} 6 & 4 \\ 4 & 6 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 18 \\ 18 \end{bmatrix} \] Using matrix inversion or substitution, we find: \[ x = 2.64, \quad y = 0.64 \] Thus, the most probable values of \( x \) and \( y \) are \( x = 2.64 \) and \( y = 0.64 \).