[Linear Algebra] 7. Find the complete solution x = xp + xn to the system [2 3 7 / 4 4 8 / 8 6 10] (3x3 matrix) x = [0 2 6] (3x1 matrix).

To solve the system of linear equations, we first need to write down the augmented matrix:

[2 3 7 | 0] [4 4 8 | 2] [8 6 10 | 6]

Next, we perform Gaussian elimination to bring the matrix to row-echelon form. We can start by subtracting 2 times the first row from the second row, and 4 times the first row from the third row:

[2 3 7 | 0] [0 -2 -6 | 2] [0 -6 -18 | 6]

Then, we can multiply the second row by -1/2 and add the second row to the third row:

[2 3 7 | 0] [0 1 3 | -1] [0 0 0 | 0]

Now, we can see that the system of equations is:

2x + 3y + 7z = 0 y + 3z = -1

From the second equation, we can express y in terms of z: y = -1 - 3z. Substituting this into the first equation gives:

2x - 3 - 9z + 7z = 0 2x - 3 - 2z = 0 2x = 3 + 2z x = 3/2 + z

So the general solution to the system is:

x = 3/2 + z y = -1 - 3z z = z

This is the parametric form of the solution, where z is a free variable. The solution set is a line in R^3.