The sum of three numbers is 10. The sum of twice the first number, 3 times the second number, and 4 times the third number is 27. The difference between 7 times the first number and the second number is 23. Find the three numbers first number: second number: third number:

Step 1 ：We have a system of three equations that we can solve using linear algebra. The equations are: \(x + y + z = 10\), \(2x + 3y + 4z = 27\), and \(7x - y = 23\).

Step 2 ：We can represent these equations in matrix form as: \[\begin{bmatrix} 1 & 1 & 1 \\ 2 & 3 & 4 \\ 7 & -1 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 10 \\ 27 \\ 23 \end{bmatrix}\]

Step 3 ：We can solve this system of equations using linear algebra.

Step 4 ：The solution to the system of equations is \(\boxed{(3, 1, 6)}\).