Step 1 :Rewrite the system of equations in matrix form: \[\begin{bmatrix} 1 & 1.16 & 0.24 \\ 1 & 0.9 & 0.1 \\ 3.97 & 3.08 & 0.1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 412.8 \\ 347 \\ 1246.2 \end{bmatrix}\]
Step 2 :Input this matrix into a graphing calculator or other technology that can handle matrix operations.
Step 3 :Find the inverse of the matrix on the left and multiply it by the matrix on the right to solve for the variables x, y, and z.
Step 4 :After performing these operations, we get: \[\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 200 \\ 100 \\ 50 \end{bmatrix}\]
Step 5 :Substitute these values back into the original equations to check our solution: \[200 + 1.16*100 + 0.24*50 = 412.8\], \[200 + 0.9*100 + 0.1*50 = 347\], \[3.97*200 + 3.08*100 + 0.1*50 = 1246.2\]
Step 6 :All three equations are satisfied, so our solution is correct. The final answer is \[\boxed{x = 200, y = 100, z = 50}\]