The least squares solution of the system of equations x+y=1, 2x+4y=5 and −x+y=−2 is

The given system of equations is:

1. x + y = 1
2. 2x + 4y = 5
3. -x + y = -2

This system of equations is overdetermined, meaning there are more equations than unknowns. In such cases, an exact solution may not exist. The least squares method is used to find the approximate solution that minimizes the sum of the squares of the residuals.

The least squares solution is given by the solution of the normal equations, which is (X^T X)β = X^T y, where X is the matrix of predictors, y is the response vector, and β is the vector of coefficients.

First, we write the system of equations in matrix form, AX = B:

A = [[1, 1], [2, 4], [-1, 1]] X = [[x], [y]] B = [[1], [5], [-2]]

The normal equations are (A^T A)X = A^T B.

Calculate A^T A and A^T B:

A^T A = [[6, 6], [6, 18]] A^T B = [[7], [7]]

Solving this system of equations gives the least squares solution.

However, the matrix A^T A is singular (its determinant is 0), so it cannot be inverted, and the system of equations (A^T A)X = A^T B does not have a unique solution.

This means that the original system of equations does not have a unique least squares solution. The solution will be a line or plane in the x-y space, rather than a unique point.