7. Find the complete solution x = x_p + x_n to the system [ 2 3 7 / 4 4 8 / 8 6 10 ] (a 3x3 matrix) x = [ 0 2 6 ] (a 3x1 matrix).

To solve the system of linear equations, we first need to write down the system in the form of Ax = b, where A is the coefficient matrix, x is the vector of variables, and b is the constant vector.

Here, A = [ 2 3 7 / 4 4 8 / 8 6 10 ], x = [ x1, x2, x3 ]^T, and b = [ 0, 2, 6 ]^T.

Step 1: Check if the matrix A is invertible The matrix A is invertible if its determinant is not equal to zero. The determinant of A can be calculated as follows:

det(A) = 2*(410 - 86) - 3*(48 - 410) + 7*(46 - 48) = 2*(-8) - 3*(-8) + 7*(-8) = -16 + 24 - 56 = -48

Since the determinant of A is not equal to zero, the matrix A is invertible.

Step 2: Find the inverse of A The inverse of A, denoted as A^-1, can be found by using the formula A^-1 = adj(A)/det(A), where adj(A) is the adjugate of A. The adjugate of A can be found by taking the transpose of the cofactor matrix of A.

Step 3: Solve for x Once we have A^-1, we can solve for x by multiplying both sides of the equation Ax = b by A^-1. This gives us x = A^-1 * b.

Step 4: Check the solution To check if the solution is correct, we can substitute x back into the original equation Ax = b. If both sides of the equation are equal, then the solution is correct.

Note: The above steps are a general approach to solving a system of linear equations. Depending on the specific values in the matrix A and the vector b, there might be shortcuts or simpler methods to find the solution.